I did a web search on "coloring cotton balls" and what I found is pretty simple. Mix a solution of water and food coloring to the desired color. Other sites suggested using tea bags or KoolAid as well. Soak the plain white cotton balls in this solution for half an hour, then dry on a rack.
I tried this tonight using a dish of water (about 3/4 a cup), several drops of red food coloring, and I added about a tablespoon of vinegar only because I've used vinegar to set dyes in the past. I soaked half a dozen cotton balls in it for 15 minutes. Then I let them drain on paper towels for a few minutes, rolling them around a little, and popped them into the clothes dryer on "medium." They came out red and looking pretty decent. Not pastel, however, but that may be from using too much food coloring.
Toothpicks came in handy for picking the wet balls out of the food coloring.
I thought about using rubbing alcohol instead of water, too, since it would dry faster, but if you try that, don't use your clothes dryer!