As anyone who ever plays tabletop games online with me can attest, I have three rambunctious daughters. Two of them are of an age that I am trying to get them into tabletop gaming (5 and 7 years old.) Unfortunately, my GM style is more prone to the combat and creepy elements of play, so it is a bit of a learning curve for all parties.
The system I whipped up for them seems to work quite well for their age range so far: a d20 roll, with a target number of 20 and three skills (Befriend, Work, and Fight.) If I need to make something easier or harder, I halve or double the value for the skill they are using, which is introducing the children to fractions and multiplying, albeit in very simple ways.
The setting has all three players (including my wife) as princesses, tackling problems their father has in his (small) kingdom. So far they have recovered animals kidnapped by goblins, fought goblins to save a church, and helped free some people turned into goblins by an evil, shape-shifting, army-summoning witch. Cinderella (no relation to the fairy tale character) already wants to marry the handsome prince who is the cured leader of the people-turned-into-goblins.
Anyway, that is what I have been doing to introduce my girls to roleplaying games, when I am not working on writing my own. I am always interested to hear what other people have done to introduce kids to roleplaying games.