I'd posted earlier about live roleplaying with kids, and I've just started looking into systems for tabletop roleplaying with kids. My daughter is just about six months old now, so I have some time before she'll be ready, but I'm pretty excited about the idea.
I found a great system called Shadows that looks like it'll work great with kids. The basic idea is that each person has a good side and a bad side (the shadow). The good side wants to do good, and the bad side wants to cause trouble. So whenenver a character wants to do something in the game, the player has to say what the character wants to have happen, and what the character's shadow wants to have happen. They roll a six sided die for the good side and a six sided die for the shadow, and the highest roll tells you which side won.
Shadows allows a lot of player input into the story. For example, if a character hears a noise in the fireplace and goes to investigate, their good side outcome could be, "Santa appears", and their shadow's outcome could be, "Evil elves appear and steal the presents". From that point on, whichever outcome happens will affect the ongoing storyline.
While that might be a bit odd for people used to more traditional roleplaying games (e.g. Call of Cthulhu, D&D), it's great for kids because it encourages them to be creative and to think about the possible outcomes of decisions. Wonderful stuff!
You can find Shadows available for free at http://www.harlekin-maus.com/games/shadows/shadows.html.
If you're into roleplaying with kids, I also highly recommend the Kids RPG mailing list, at http://www.samchupp.com/kids-rpg.html. Lots of people there with great advice for introducing kids to roleplaying.