One of my big interests with role playing games, besides playing them myself, is encouraging parents to role play with their kids. Role playing with kids is a great way for parents to really find out how their kids react to failure, to loss, to success, and to give the kids a way to see the consequences of decisions without actually suffering those consequences (except by proxy, through their characters).
One of the absolute best role playing games to start with is Shadows, by Zak Arntson. As a bonus, it's a free game available on the web.
Shadows has a simple system, so if your child can count to six, and compare two numbers to see which is higher, they can play Shadows. And if they cannot, they'll still enjoy rolling the dice and you can tell them the result.
The basic idea behind Shadows is that, for every significant action a character wants to take, the player must specify two outcomes. One is what the character wants to have happen, the other is what the character's shadow wants to have happen. The character's shadow is an invisible person that wants to get the character in trouble. The shadow doesn't have any real existence in the game, other than to serve as the inspiration for the "bad" outcomes.
Once the player announces the two outcomes, they roll two standard six sided dice. They need to be different colors. Black and white would be traditional, black for the shadow outcome and white for the good outcome. If the good die is equal to or higher than the shadow die, the good outcome happens. If the shadow die is higher, the bad outcome happens.
There's also a nice little mechanism to allow players to affect the dice rolls of other players. Each player starts with three tokens, and can spend a token to force a player to reroll either their good die or their shadow die. When you spend a token, you give it to the player you're asking to reroll. This can be used to help them, for example when their good die is lower than their shadow die, or to annoy them by making them reroll a high good die.
The dice rolling isn't intended to get in the way. The GM can feel free to simply narrate what happens until rolling dice would increase the tension. So you don't roll dice when the player says their character is getting out of bed, but you do when they say they want to stop a thief from running out of the house.
Shadows is a very simple game that encourages a flowing story line with liberal contributions from players. Great for kids, who already play like that anyway when they're pretending. Games can be serious or light hearted, whatever everyone feels like playing.
So the next time you're wondering what to do with your kids, break open a board game that uses six sided dice, and give Shadows a try.