“Imagine a 64-square chessboard with a coin on each square.
Each of the coins has either head or tails facing up, chosen at random.
I check the board and decide which coin will be my favorite one.
One of you (say A) will be with me, see the chessboard and I will reveal to him (only to him) which coin is my favorite. He then must flip over exactly one of the coins on the chessboard, while the other mathematician (B) is in another room not looking.

Once the coin is flipped over, the uninformed mathematician (B) is summoned into the room and must deduce which coin is my favorite only by examining the coins on the chessboard.
To secure absence of any other hints A is escorted out of the room.

Clearly, prior to the procedure, you are free to discuss the problem between the two of you and establish its solving strategy. You have no time limit, you are free to use any kind of calculator, but any communication between you two is strictly prohibited”

What strategy can the two mathematicians devise to ensure that my favorite coin can always be correctly identified?

Hamming says that you need 6 bits of data to identify a one bit error (the 'favourite number') in an 8*8 matrix. It's basically the same idea as the 'guess my number' magic cards I had as a kid.

Detecting the error and choosing the card are essentially the same, just different ways of looking at the same thing.

In that case,  my question is, why bother with Hamming at all? Your proposition is equivalent to saying that A+B can agree some arbitrary identifier (it might as well be 000000) and then automatically detect a one bit error in an 8*8 message. Hamming doesn't come into it, and indeed if true it would contradict Hamming.