A fish is another classic first pet, provided you pick the right one. Goldfish are the archetypal choice, but they're notoriously fragile and require a fairly elaborate tank-and-filter setup.
Betta fish, however, are happiest in smaller bowls, no filter necessary. Bettas are beautiful fish, often jewel-toned, with long flowing fins. The bowl will need regular cleaning and water changes; consider adding an aquatic snail to your bowl, which will help keep the algae at bay. (They're fun to watch, too.) Your child can have fun decorating the bowl with gravel, plants and other accessories. A word to the wise: This is one pet that's best purchased singly, because two bettas in the same bowl will fight to the death. But a healthy, well-tended betta can live for two or three years.