There are quite a few theories floating around out there, but as it's not a universal standard and mainly contained within Western Europe and cultures influenced by Western Europe, I'm inclined to go with the tick / cross system being distributed by the Romans, especially given the amount of bureaucracy they bought to the world.

Common theories:

The Roman "Veritas" (truth) being the origin of the tick, cross is just a handy symbol / easy to make with a stylus / origin unknown.

The Greek "Nikas" (νικώ) for win and "Haneis" (Χάνεις) for lose.

I personally like the Greek theory as it accounts for the cross and the Romans did appropriate an awful lot from the Greeks. But that's 100% conjecture.

Be interesting to see if any of the real historians out there have access to Roman or Greek documentation :)

In Italy crosses represent correct answers. In questionnaires we put a Xs beside the correct choice and, when casting our vote during elections, we cross with a giant X the party symbol we want to choose.

So check marks did not come to represent 'right' while Xs represent 'wrong' everywhere.