I am a clay artist so I can offer you a couple options...
1) The easiest for a "non artsy person" is to go to a tile outlet and buy some white (or whatever color) tiles. Unglazed is best, but usually not available. Then you can go to a craft store and buy regular acrylic paints in the small bottles. They come in every color, and are about $1.00 for a 2 ounze bottle. They also sell a clear, craft spray paint. Like a varnish. So the kids would paint what they want w/ the colors, and you would then go back later and spray it w/ clear. You can also use 1/2 an egg carton to divide up the colors between kids. Each one gets a squirt of 4-6 different colors. (Otherwise they mix them all together!) We did this once w/ my sons Kindergarten class. It was awhile ago and I can't remember how long they took to dry,sorry. And the clear turned yellow a few months later. Some people don't care.
2) I believe it was in Michael's craft store I saw a paint for painting piggy banks. It was either air dry, or something you may have to put in the oven to cure. You can look into that too. Same idea- buy the tiles, and then use that paint on them. Personally, I would not put anything like that in my oven that I cook my families dinner in. Some people aren't bothered by it, but all that stuff gives off fumes. We breathe enough bad stuff in everyday, I don't want to add more.
3)If you want to do it right, this is what to do:
Find your local ceramics store/or paint your own pottery store. Go there, tell them what you are doing and ask if they can fire these tiles for you. If so- you'll buy unglazed tiles for the amount of kids, and extra in case parents want to try. Buy a few 2 ounze bottles of what is called "stroke n coat" or Duncan's brand is "concepts." They come in many different colors. You will also need a bottle of clear.Do the same thing I said above, give the kids a couple colors in an egg carton. Let it dry completely. Then dab the clear over top. DO NOT BRUSH ON THE CLEAR. Otherwise you'll rehydrate the colored glaze underneath and it will smear. Use a thick brush and DAB it. This store should also be able to sell you a pencil that won't burn off in the firing. You can write the kids names on the back of the tiles before they start. Because you won't remember later! You'll then need to take these back to the ceramic store to fire them. This way is going to be more expensive than the first 2 options- but this is the right way. Then you can always invite each kid over at another time for a play date and to pick up thier tile.
Email me if you have more questions. If you live in St.Pete, I could tell you where to buy these things. I might even be able to help you myself, and do the firing. Let me know.