MATL, 18 bytes

ZBtE:t2/kZ~tb=fQ)B

Try it online! Or verify all test cases.

Explanation

Let a(n) denote the sequence of integers corresponding to Gray codes (OEIS A003188). The program uses the characterization a(n) = n XOR floor(n/2), where XOR is bit-wise.

Essentially, the code converts the input to an integer a₀, finds that integer in the sequence, and then selects the next term. This requires generating a sufficiently large number of terms of the sequence a(n). It turns out that 2·a₀ is sufficiently large. This stems from the fact that the Gray code a(n) never has more binary digits than n.

Let's take input '101' as an example.

ZB      % Input string implicitly. Convert from binary string to integer
        %   STACK: 5
t       % Duplicate
        %   STACK: 5, 5
E       % Multiply by 2. This is the number of terms we'll generate from the sequence
        %   STACK: 5, 10
:       % Range
        %   STACK: 5, [1 2 3 4 5 6 7 8 9 10]
t       % Duplicate
        %   STACK: 5, [1 2 3 4 5 6 7 8 9 10], [1 2 3 4 5 6 7 8 9 10]
2/k     % Divide by 2 and round down, element-wise
        %   STACK: 5, [1 2 3 4 5 6 7 8 9 10], [0 1 1 2 2 3 3 4 4 5]
Z~      % Bit-wise XOR, element-wise
        %   STACK: 5, [1 3 2 6 7 5 4 12 13 15]
t       % Duplicate
        %   STACK: 5, [1 3 2 6 7 5 4 12 13 15], [1 3 2 6 7 5 4 12 13 15]
b       % Bubble up
        %   STACK: [1 3 2 6 7 5 4 12 13 15], [1 3 2 6 7 5 4 12 13 15], 5
=       % Equality test, element-wise
        %   STACK: [1 3 2 6 7 5 4 12 13 15], [0 0 0 0 0 1 0 0 0 0]
f       % Find: yield (1-based) index of nonzero values (here there's only one)
        %   STACK: [1 3 2 6 7 5 4 12 13 15], 6
Q       % Increase by 1
        %   STACK: [1 3 2 6 7 5 4 12 13 15], 7
)       % Apply as index
        %   STACK: 4
B       % Convert to binary array
        %   STACK: [1 0 0]
        % Implicitly display

Jelly, 10 8 bytes

Thanks to Dennis for saving 2 bytes.

^\Ḅ‘^H$B

Input and output are lists of 0s and 1s.

Try it online!

Explanation

The inverse of the Gray code is given by A006068. Using this we don't need to generate a large number of Gray codes to look up the input. One classification of this sequence given on OEIS is this:

a(n) = n XOR [n/2] XOR [n/4] XOR [n/8] ...

Where the [] are floor brackets. Consider the example of 44 whose binary representation is 101100. Dividing by 2 and flooring is just a right-shift, chopping off the least-significant bit. So we're trying to XOR the following numbers

1 0 1 1 0 0
  1 0 1 1 0
    1 0 1 1
      1 0 1
        1 0
          1

Notice that the nth column contains the first n bits. Hence, this formula can be computed trivially on the binary input as the cumulative reduction of XOR over the list (which basically applies XOR to each prefix of the list and gives us a list of the results).

This gives us a simple way to invert the Gray code. Afterwards, we just increment the result and convert it back to the Gray code. For the latter step we use the following definition:

a(n) = n XOR floor(n/2)

Thankfully, Jelly seems to floor the inputs automatically when trying to XOR them. Anyway, here is the code:

^\          Cumulative reduce of XOR over the input.
  Ḅ         Convert binary list to integer.
   ‘        Increment.
    ^H$     XOR with half of itself.
       B    Convert integer to binary list.

Haskell, 118 115 bytes

g 0=[""]
g n|a<-g$n-1=map('0':)a++map('1':)(reverse a)
p=map(snd.span(=='0')).g
f s=(snd.span(/=s)$p$1+length s)!!1

Try it on Ideone.
First approach: Generate the set of all gray codes with length(input)+1, remove all elements until input is found, return the next element.

JavaScript (ES6), 58 bytes

s=>s.replace(s.split`1`.length%2?/.$/:/.?(?=10*$)/,c=>1-c)

Directly toggles the appropriate bit.