The Third String

Haskell, 43 bytes

x!y=[s|s<-(<$max x y)<$>"abc",s/=x,s/=y]!!0

Takes the max (lexicographically later) string, which we know is nonempty; replaces all characters with one of "a", "b", and "c" using <$; and returns the first that is neither of the inputs. I think this is similar to Neil's Charcoal answer and/or geokavel's CJam answer.

(I've lurked for a while but this is my first time answering on this site; hi!)

Brainfuck, 97 bytes

+>+[-,[>>,]<<[<<]>]>>[>]<<<[<]>[<+>>[-<+<+>>][>]<[->+<]<[-]>>>++<[<]>[.>>]>[>>]]>[+[>]+[<]>[.>>]]

Run code online (note that "dynamic memory" must be selected in the bottom-right)

Awesome challenge! I thought it would be trivial but it ended up being really difficult. I keep coming back to it because I feel like there should be some elegant 20-or-so-byte BF solution. At this point, I'm pretty happy I (seemingly) got it to work at all in BF.

Input is taken as str1 + \0 + str2, where strings are consecutive non-zero 1-byte characters.

Returns (first str1 + first str2) or (first str1 + 1) or 2. This algorithm was thought up by the brilliant @ØrjanJohansen, (presumably) based on my (broken) original one.

Commented:

# Let (Kn) be the nth character of K
# Let (^) designate the pointer
# Let F be the first string inputted
# Let S be the second string inputted

+>+[-  # Twice do
,[>>,]  # Input string (characters separated by 1)
<<[<<]>  # Go to left of beginning of string
]>  # End on first character of second string
# If second string is null we will end one too far to the left
>[>]<<<[<]>  # If first string is null we will end there too
# We will use this to do flow control

[  # Only run if both strings were non null

# Tape:    S0 ' F0 ' S1 ' F1 ' S2 ' F2 ' etc
#          ^

<+>>  # Let F0* = F0 (plus) 1  (is 1 as of now; F0 will be added later)
[-<+<+>>] # Let A = S0 (plus) F0
# A may or may not be zero
# F0* may or may not be zero
# Tape:    F0* ' A ' 0  ' S1 ' F1 ' etc
#                ^

[>]<[->+<]  # Let B = A or F0*
# B may or may not be zero
<[-]>>  # Clear F0*
# Tape:     0 ' B ' 0 ' S1 ' F1 ' etc    (if A was zero)
#               ^
# OR        0 ' 0 ' B ' s1 ' F1 ' etc    (if A was nonzero)
#                   ^

# Let C = B or 2
# C will be guaranteed nonzero and unique from S0 and F0
>++<[<]>  # Create C
[.>>]  # Print (using S or F; does not matter)

>[>>]  # End on a zero cells with zero cells all to the right
# This is necessary for the coming functionality
# also as to not loop
]  # End non null block

# Now we consider if one of the strings was null
# Tape:    0 ' E0 ' 0 ' E1 ' etc    (if one string was null)
#          ^
# Tape:    0 '  0 ' 0 '  0 ' etc    (if neither string was null)
#          ^
# Where E is F or S (we don't care)

>[  # Execute only if one string was null

+  # Let A = E0 (plus) 1
# A may or many not be zero
# Tape: 0 ' A ' 0 ' E1 ' etc
#           ^

[>]+[<]>  # Let B = A or 1
# B is guaranteed nonzero and != E0
# Tape: 0 ' B ' ? ' E1 ' 0 ' E2 ' etc
#           ^

[.>>]  # Print

# End on zero cell as not to loop
]  # End null block

Jelly, 8 bytes

żḟ@€ØAZḢ

Try it online!

How?

żḟ@€ØAZḢ - Link: list of characters, a; list of characters, b
ż        - zip a and b
    ØA   - uppercase alphabet
 ḟ@€     - filter discard for €ach (swap @rguments)
      Z  - transpose the result
       Ḣ - head