Old beach towels are the best. The larger the towel the better, although for young children, probably a regular size bath towel is all that they will be able to work with easily. All you need for cleaning solution is warm water. If your tile is really dirty, you might put a drop or two of dish soap to a gallon of warm water. Use very little if you use any at all.
Warm water works the best for most jobs.
Make sure the water temperature isn't too hot if you have children helping you Wet and ring out your towels. This is the hardest part, but it can even be fun if you do it in the bathtub. Have a child stand in each end of a dry tub. Have each child take an end of the wet towel. Each child needs a towel to start the fun of cleaning the tile.