Pre-Literate Children: Before a child can read, they can still do imaginative play...and this is where you get started. Assist the child in her make-believe, create castles, spaceships, old west corrals. Do what you can to encourage role-playing activity. This is the time when you start to instill a sense of adventure and creative expression. Of course, you're going to have to get in there and RP, too! Your child will feel even less inhibited if you're willing to interact and model good RP.
This might be a good time to model "Rules" as well - at least, as rough guidelines. Using a modified rock-paper-scissors challenge system might be good enough to solve the "I GOT YOU! NO YOU DIDN'T! YES I DID!" arguements. Encourage character development by asking questions about the character your child is playing, such as:
"What does she eat for dinner?"
"Who are her favorite people?"
"What does she like to do for fun?"
"Does she have any friends? If so, who?"
This will help the child start to think of the *make believe* persona as a character, with independent thoughts, hopes, dreams, and goals...which will also be handy for differentiating between "magical thinking" and RL.
Post-Literate Children: Kids who can read can play your basic everyday tabletop RPG. The trick is to find one that they like. The trick to finding one they like is to find character concepts that excite them. The trick to finding exciting character concepts is to watch how they react to various characters in the media or in stories you tell.
For example, Jay, a young man in my kids' gaming group, is kind of a hipster sort of kid who likes to be cool. He thinks Keanu Reeves' character Neo in "The Matrix" is pretty cool. So I gave him a character who looked and acted and seemed like that character...and I think he really likes it.
Gee, another gamer in my group, likes to play faeries and such. She enjoys playing a faerie dragon. I had to make up my very own core character class, the "Faerie Dragon" class, to allow her to play and advance with the rest of the players, but so far it has really worked to get and keep her attention.
The point is, don't quibble with the child. Just create the character they want, the character of their dreams: even if it's a little silly, it will pay off when they catch the "gaming bug" and start *asking* you to play.
Tweens & Teens
Tweens (ages 10-12) and Teens (13+) must be approached differently. My thought is that you would do better to take a clearly cool "adult" RPG and tone it down in terms of hardcore "mature" content and offer it to the teens and tweens to play. They'll feel more grown-up and in the meantime you can still share a common language and common experience gaming with them.
Whatever you do, remember: kids are not tiny adults. But they do not wish to be looked down upon. They are people who have very little power and are constantly having to worry about what they may or may not do. Being someone else for a while can give them the opportunity to have fun and make mistakes in a safe way, without causing them lasting harm.
Thanks for listening!