As this is a very unrealistic scenario I tried to think of any realistic answer that would handle all of the variables. My best response is that the projectile would pierce the case of a computer merely as a delivery platform for specialized nanites, programmed to form virtual networks and launch various penetration attempts against the system and its components. It's possible that the projectile could serve other functions as well, such as providing a wireless power source for the ninites, or providing a wireless communications channel through a shielded computer case. How much of this could be accomplished with existing technology I do not know, and there are likely more useful methods for accomplishing such things (see social engineering).

This requires a bit of setup, but I think it works.

Early in the story, have engineers discuss that the new computer architecture really isn't all that fault tolerant. If anything goes wrong, it pretty much has one fail mode: brick (do nothing). Even non-critical systems trigger the fail mode. The exception is when there's a debugger attached by wireless communication, which allows tech support to diagnose the issue and do things like upload driver patches. I've seen some devices that work like this. Having a full computer work that way is improbable, but an embedded computer, like a bridge/dam monitor would be unsurprising for me.

Later, your hardware assassin (HA) creates a gun like a nail gun, with a payload. I'm thinking something like one of those toy rockets from the 1950s with the super sharp points on the tip, without the fins. The body of the rocket has wireless device in it, like a Raspberry Pi.

HA fires the nail gun at the target computer, aiming to hit an unused USB port. Puncturing the port may create a device error since those ports are polled regularly for a connection. That's the only non-critical system I can come up with that is targetable and active, particularly on the embedded targets I mentioned earlier. HA probably has a short range of a few feet.

The target goes into fault mode. The wireless in the rocket body opens communication and says, "Hi. I'm a debugger. I'm here to help you." It proceeds to patch, likely with admin access, critical code. And then it does whatever it needs to do.

This idea skirts the edge of believability, I admit, but perhaps it is a shell you can build from.

It may not be as silly as you think.

Sound based attacks on encryption have been demonstrated, and other processing tasks might leak other data. Observation and potentially control of signals on wires is possible at short ranges. Some chips are now using wireless communication to avoid issues with moving low power high speed data over small wires.

Getting inside the case would make these less silly-sounding even though you don't need to with the real world attacks. The hacking part probably needs to be targeted to a device, so the projectile might be too. Either using cameras and control surfaces to aim itself at a known safe spot in the case, or maybe having a well designed shaped charge that opens a hole without throwing around much extra energy to damage things and a soft part of the projectile going in at a reasonable speed.

Redundancy and failsafe are expected to be words computer users like to hear for the foreseeable future. High end systems (specially the ones that might be expected to get hit by things) try to still work when some damage is done, quite possibly by switching to identical backup parts in the same or a nearby box.

I can think of a couple, admittedly far fetched, scenarios where a "hacking arrow" might be useful.

1. PCs are protected by a faraday cage. So the arrow head pierces the case in order to breach the cage and be able to wirelessly interact with the sensitive non-hardened components inside. This would presume that PCs rely almost exclusively on the cage for protection from remote hacking, once you breech the cage it is easy to hack the PC with a Bluetooth, NFC, or Wi-Fi transmitter.

2. The arrowhead is cutting a CABLE, essentially bridging the gap and allowing a microcomputer inside the arrow head to read all the data being transmitted by the cable and insert it's own code. So imagine a data cable that is being physically severed but the arrowhead still passes info to each severed end. Upstream and downstream PCs may not notice a difference (perhaps slight lag?). Of course this can be done in the real world and doesn't even need to physically cut the cable. Obviously extraordinary marksmanship would be required.

3. The arrow head has a powerful transmitter that needs to be very close to the target in order to overpower electronic countermeasures/jamming. Sure, the building may be secure from intruding transmissions, but if you could deliver a transmitter RIGHT NEXT to the target, it could cut through the interference and "hack" the target.

I don't think that a piercing projectile can be of any use here, maybe something sticky that glues itself to computer device, melts a hole and sends in nanobots or something like that.

Just think for a second about shooting a metal box with an arrow. In order to pierce you need to shoot at a certain angle, with great power and there are also many other things you need to consider before getting to the data transfer problem, because in reality arrows are by far not the best projectiles.

If you really want a projectile that pierces I'd go with a bullet, but again, what if you damage some physical connections that are in the device and won't be able to transfer data? Or damage the components that store data, or short circuit will happen and instead of any hacking it will just burn the device. Too many possibilities for a bad ending. It's unpractical.

Generally speaking, if such thing was possible, it would exist in our world by now. To answer your question we need more information about the level of computer technology that exists in your world. It's like the current level of tech on Earth, then it's impossible to have hacking bullets.

You mentioned XCOM series, but I don't remember anything like that there, beside hacking with a flying robot that connects to something, maybe you mean EMP grenades? Though it's not hacking.

Bridging the air gap

First why are you shooting projectiles to hack?

Well it must mean that you have limited / no access to the system and you are using this odd tool to open a new attack vector.
The simple version of this is an air gapped or firewalled system. It is also true that some newer operating systems offer different authentication methods for a local user rather than remote (camera authentication is now available) these might be easier to attack then the modes accessible through the internet.

How does the projectile interface with the machine?

This is the hard part computer parts and ports tend to be small enough so a miss by more than a millimeter will make the system fail. So you need to be freakishly accurate but also "hit" lightly enough to not damage the port or component you are attaching to.
The first idea that springs to mind is to have a frame the pads the landing and attaches to the outside of machine the frame then detects the location of the port (usb or whatever you desire) and connect to it. This gives you a stationary and stable platform to preform the delicate connection.
The added weight of the frame and the high accuracy requirement make this a better candidate for launch by computer sighted launcher than a human archer ... unless the archer has the super ability of ludicrous accuracy.

Does it have to actually pierce the computer, instead of just sticking to the side? If it can just stick to the side (thus not damaging anything), the arrow could use powerful antennas and hyper-advanced beamforming techniques to induce currents in specific parts of the circuitry, thus changing the data as desired.

Given that most computers have little space between the components and the case (including a desktop tower on its right side), I think the best approach would be to stick to the side with an instant adhesive. The metal case would be the greatest issue, as it interferes with the electromagnetic hacking, but you can probably write it off as a problem solved by the hyper-advanced beamforming.

Edit: As for the instant adhesive, it could work like how Post-Its work, by having zillions of tiny glue bubbles that are popped on contact, causing them to stick. Of course, something stronger than Post-It glue will be required.

You're definitely getting into the fictional realm of science fiction here, but let's examine the variety of computer bits that you could hack, and some of the reasons that might (or probably wouldn't) work. I think I'll go from hardest/most valuable to easiest/least valuable.

Processor/Motherboard/BIOS

The most valuable target in your computer is the processor. If you could actually capture the signals that are going in and out of the CPU, you could potentially setup an emulator to replay the data. Additionally, you could inject your own signals into the stream (bonus points if you can sever the traces on the mother board and become a Man in the Middle (MitM)). Then you could alter the signals however you want.

Other points on the motherboard offer you similar kinds of benefits, like the RAM.

The problem, of course, is that you're talking about teeny tiny traces. Also, sensitive voltages. It's pretty likely that you'll crash the system when you try this. But you could do it.

If you can overwrite the BIOS with your own, however, you've also taken over the system. Though a lot of newer systems actually have certificates baked into the hardware, so monitoring would be a better thing here.

Disk drives

You can pretty much apply the same principles to disk drives - monitoring and injecting. The trickier business with the hard disk is that it's likely for a high-value target that by the time the bits make it to the disk they've been encrypted by the OS that's running in RAM.

Ports

You could probably reasonably develop a system with current technology that would allow you to fire darts into unused ports. Naturally they can't be blocked, but you could do something to that effect. Or perhaps your hacking tool has the mechanics to unplug something and then become a MitM.

Probably one of the most effective usages here would be attacking an airgapped target. Of course the unfortunate (for your attacker) thing about an airgapped target is that it's probably going to be pretty obvious that it's being attacked.

External devices

Depending on the circumstances and the device there may be an opportunity to attack external devices. Keyboards are probably the most valuable. For network cards, again by the time the information reaches the card it's probably going to be encrypted - especially in a high-value target.

Pierce and Play

So this is going to be a b it of a stretch, but bear with me. To start with we need to think of the projectile as two parts; the inside is going to be a high-tech computer of sorts, and the outside will be an incredibly strong and well-designed conductor.

The Shell

This is the part that is going to be hardest to swallow, but I'll give it the old college try. The outer surface of your projectile needs to be made of some kind of super conductive material. The idea is that after you physically insert it into the system that you are trying to hack, it will be able to essentially replace whatever bits of the original architecture it displaced with itself. Data is really just electricity moving around, so if your projectile can continue to move that electricity in the same way as the original hardware it shouldn't be a huge interruption to the computer.

Of course, for this to work out you are going to want your projectile to be incredibly sharp and as small as possible, to destroy the least amount of physical hardware. Something like a crazy high tech dart might be the best option for this.

The Guts

The inside of the projectile is just as fantastic and unlikely, because what you need is a computer that can figure out what kind of data is being sent through the projectile and then alter it in some way. Think of it like a man in the middle attack, only the man is actually an arrow (or dart, or whatever) and what he is in the middle of is your computer.

Handwaving at the Highest Level

So, lets pretend that you have this kind of projectile and go to use it. You throw a dart at the target computer, it pierces into it and begins acting as a conductor for the physical parts of the computer that it just destroyed. The onboard computer analyzes the data that is passing through the projectile and then alters it in whatever way it is programmed to. Eventually this leads to the targeted computer getting "hacked" for whatever definition you want to use. Once you are done you can remove the projectile, which will break that connection and kill the computer it was shot at.

Solid Snake

Why not have the shaft of the arrow contain a snake-like robot which finds and plugs itself into any unused 1394/USB headers on the mother board. The tip of the snake is therefore just the header with a wire to the base of the shaft where the controller sits with a small camera for identifying the pattern. At that point you have physical access and unless the computer admin has taken specific action to disable the unused header, you basically own most computers. The shaft also contains batteries (until it can charge from USB) and acts as an antenna for wireless transmission.

So this is a way out there... but considering the examples all come from soft sci-fi sources maybe it'll be ok.

So first the head of the arrow needs to be able to pierce thin steel, but not penetrate too deep, so a sharp hard point with arms coming out the side to let just the tip in, and keep it from going right through to the sensitive stuff.

Behind the head and maybe up into the shaft of the arrow would be electronics, with a bunch of little super thin wires that would extend, seek out and tap into key areas like the ram connections, processor, hard drive, etc.

Well this answer admittedly doesn't answer the question as is because it doesn't fit many of the requirements. But I think it is a bit more realistic so could be of use for ideas.

Frozen RAM bypasses all disk encryption methods

Encrypting a hard drive can be a great way to ensure data stored on a computer is safe. To access this data you type in a password and it is stored in RAM so that the computer can continue to read/write to memory. This is considered safe because RAM is volatile so no one can come in and read that password out of memory. However, it has been shown that if you greatly reduce the temperature (freeze) RAM it drains its memory much much slower, thus giving an attacker time to plug the memory into their own machine and read the encryption key.

So, if you want to work this into a story, you can detail a room with top secret encrypted computers. The hacker has devised a way to gain access to the room but as soon as he does an alarm will trigger locking down everything and shutting down the computer. To combat this, before entering the hacker shoots an arrow into the computer which then delivers a payload of liquid nitrogen. He then enters the room, grabs the frozen RAM and encrypted hard drive, and makes off with all the tools needed to get to the secrets.

Not as flashy as remote downloading arrows like in the avengers but real life never is.

Make up computers vulnerable to such attacks but resistant to physical damage:

Our present day computers are very likely to crash if you broke their casing and start fingering their tiny parts. But take the human brain for example: It is much more sensitive to many factors, for example pH change or temperature, but can endure quite a much localized damage without shutting down and even learn to substitute the damaged parts.

Make a computer which has many independent but connected physical parts processing information simultaneously. (Not a classic computer build around a processor and running a neural network, but real, hardware based artificial neurons.) Such a device won't be used as PC or data storage, but to control something (robot, drone swarm...) which has to intelligently react to it's environment. If you breach it's casing and wipe out some thousands neurons, it won't wreck, but will start to adapt to the situation and reroute its computing to unharmed parts. But your projectile will connect to the places of the broken neurons and send in false information. If you know enough about the system you hit, it may will be able to take control over the device.

Like most of the answers, this requires some handwaving:

Let's assume your target computer is connected to a standard keyboard by an unshielded wire. You fire your hacking arrow into the keyboard (or maybe at the wire itself). It outputs an electromagnetic signal powerful enough to cause interference in the unshielded wire, generating signals that get sent along the wire into the computer. If your keyboard uses a standard protocol, then the hacking device can output signals that get transmitted along the wire and emulate keystrokes.

Now your arrow can 'type' input into the target system. If the operating system is known, you come up with a standard set of keyboard commands that should perform a specific action on the target machine, such as (on Windows)

[windows key] + [R]   //opens 'run' dialog
[c][m][d] [return] //opens command prompt
[e][r][a][s][e][ ][c][:][\][w][i][n][d][o][w][s][\][*][.][*] [return]
[y] //confirm operation

I believe that this is possible, and it can be even more discreet/fast than you think. You state that the required "software" is set up and that is needed in my idea. Say you have a RFID scanner setup. Then you shoot an arrow with an RFID chip inside that has your hacking codes or whatever into the computer/receiver. If the RFID scanner continues to be able to actaully pick up the signal you send it will activate the commadns you sent. Now probably if you want it to look like it "hit" the machine, then make the computer and arow head magnetic. That way, hopefully with no penertration, the info is delviered.

Better yet, you could just have the arrow with the information whiz above the reciver and delvier the commands.

I think hacker arrows make sense with one condition:

The computers have hardware backdoors, either by poor design or sabotage

Say that the computers have hardware installed that can open or shut a programmatic backdoor (an obviously bad idea, but a surprisingly common one).

Or maybe you know a rebel scientist who works in chip manufacturing and has created hidden backdoors. If you can fry that hardware the right way you'll open the backdoor.

What does "fry" mean? I don't know, ask the rebel scientist. Hell, he probably designed the arrows. But maybe it means something like wireless capacitor charging...

In June 2016 it was reported that University of Michigan Department of Electrical Engineering and Computer Science built a hardware backdoor that leverages "analog circuits to create a hardware attack" so that after the capacitors store up enough electricity to be fully charged, it would be switched on, to give an attacker complete access to whatever system or device − such as a PC − that contains the backdoored chip. In the study that won the "best paper" award at at IEEE Symposium on Privacy and Security they also note that microscopic hardware backdoor wouldn't be caught by practically any modern method of hardware security analysis, and could be planted by a single employee of a chip factory.

https://en.wikipedia.org/wiki/Hardware_backdoor#Examples

Mentioning this since I don't see it already.

The projectile doesn't hack, it just locates the computer

So, imagine someone like Tony Stark.

He's already got access to any machine in the world he wants through his sophisticated mainframe. The problem is identifying that specific computer over there.

The projectile projects a carrier frequency physically into the hardware that can be detected on the global internet so the super-computer can find it and break in from the outside.

With known systems and technology, I think the shooter should target the parts of a system that can be swapped while the system is running, and not attempt to break/wirelessly control components that cannot be hot-swapped if the system must keep running, because the latter would take advanced mini robots to precisely place tiny connections, at least. Wireless control of any component with the size constraint of a projectile is rather unlikely unless there are known wireless vulnerabilities, though that may be viable if you know a way to wirelessly hack the system, but the security doesn't allow anyone to set up any wireless device anywhere near it - then shooting a wireless device there might be useful, but I wonder why the security doesn't just disable the wireless receiving component beforehand in that case.

Shooting a small robot that can pierce into a network cable to decode and intercept the signals is probably easier, but if the shot must be aimed at the case, how about specific parts of a case like network/keyboard/mouse ports? Secure systems could monitor those connections, but you could imagine a determined hacker to have broken those protocols beforehand. After all, even the best computers can occasionally have a loose keyboard connection, right?

At the risk of stating the obvious, you could just have an arrow with an RJ45 plug on the end and a small single-board computer glued to the shaft. Shoot it into an unoccupied ethernet port on the front of a rack-mounted server and it connects via (say) an unsecured heartbeat network and uses some exploit to pwn the server in whatever way the story requires.

As to why you need the arrow, it's not unusual for servers to be in locked cages or for sections of data centers to be separated by welded mesh security fences (allowing servers to share airflow while keeping customers away from each other's equipment). Hitting an RJ45 would require impressive, but not inconceivable archery skill; I assume any character doing this is meant to be a bad-ass.

If you just want to shoot an arrow with flashing lights into a server and then say it's been hacked, "something something van Eck phreaking" would cover it, but I would say that is phenomenally difficult to sell if you are going into a lot of detail. Modern computers are filled with GHz-frequency components and spinning fans and bits of metal that would make it difficult even in a lab setting.