Trying to find a solution, I started putting my initials in the squares. After a while, that didn't work well, either.
Hence my current solution. I made up new charts. Punch cards, actually. Each child will get to decorate their own card with their name or picture. The cards will fit nicely in the table tubs.
You can get a copy of these punch cards here.
Now, each time they earn a punch, they'll bring me their card. Note that the hole punch I'll be using is actually a star, and not a circle like a regular hole punch produces.
We'll see how this works out.