I've been playing a slightly modified 4E game with my now-nine-year-old daughter since she was four.
The modifications are as follows:
-Experience awards and level requirements are divided by 50.
-Hit Points, damage, healing, THP and so on are all divided by 5. Variable effects such as damage use the average result, rather than rolling. Criticals do (max damage + average enhancement dice) / 5.
This keeps the math much simpler for the players -- they just need to add or subtract mostly single digit numbers.
I recommend 4E due to the great support of props (figures, tiles, and so on). We'll usually go to local gaming stores, which have their own landscape sets and gaming tables.
Some tips for running a game with younger players:
1) Use props (as mentioned). Making the game visual will make it far more immersive.
2) Work out in advance how you want pets and mounts to work in the game. The players will almost certainly want these.
3) Make sure you know the level of violence the players and their parents are comfortable with. When I started running my own game, for example, the enemies were all elementals, constructs, mindless undead, or beasts. Humanoids were strictly for talking to.
4) Consider creating a setting the players are familiar with. My own game has a "Hero School" where young adventurers go to learn their trade. Adventures are assignments (sometimes with "extra credit" side missions).