This past month, my wife talked me into running a short, simple D&D v.3.5 adventure for her and the kids. She and I did almost all the char-gen, using the 1st-level starting packages in the PH, and deliberately choosing classes without complex features or too many options (sorcerer rather than wizard, no paladins or monks, etc.). We also ignored many of the more complicated mechanics (alignment, attacks of opportunity, Concentration checks, etc.) in order to teach and run the game more quickly. It went well, and the kids want to play more someday soon.
Over the next couple days, my son decided to make up his own game, so used his (very spotty) memory of the D&D rules to work out an adventure very similar to the one I ran, but with his own touches. Partway into designing it, he asked his older sister to help out, and they played it by themselves the next day. It wasn't much more than "let's pretend" with some very vague rules attached, but they enjoyed it, he was very proud of his work, and naturally I was quite flattered. My wife and I have had few doubts that our kids would grow up to be gamers, and this pretty much seemed to clinch it!
When my wife and I were discussing what to run for them, we briefly considered True20. However, I haven't done anything with that system in ages, and my wife isn't familiar with all the rules, so we went with D&D. It's the system we know best, which made it easy to simplify the rules off the cuff.
Since then, I have acquired Unearthed Arcana, which presents generic classes and a couple different systems for damage saves that are very close to True20's approach. We'll be seriously considering using some of those rules to streamline things further, as well as taking more time to consider just switching over to True20. On the other hand, we don't want to overthink this "D&D Lite" experience, because the kids will be old enough to grasp the full d20 rules in just a few more years. Now THAT is a thought that is both scary and exciting!