Balls were very popular (usually inflated bladders). This example is Greek, but I believe the scene in the Odyssey in which Odysseus washes up on shore and Nausicaa is playing ball with her companions is the oldest mention of a ball game in western literature.

Children also commonly played with dolls and carved toys and rattles and dice. The Metropolitan Museum of Art has some ancient toys in their collection. Many museums with classical collections have examples of these toys.