The reason I vote for Nintendo is because they have kiddy games (which I love btw) such as, Super Smash Bros., Mario Sunshine, Legend of Zelda series, ect.
In fact the Wii has free online gameing and has downloadable NES, SNES, N64, and even Gamecube games useing a credit system (not sure how that works) so you don't have to pay for them. It also can play GC games and works with the GC controlers. Not to mention I love the whole motion sensing thing.
PS2 has one advantage though as both Guitar Hero and Guitar Hero 2 are both out for that console :D .
Serioulsy though for a 6 year old you cannot go wrong with a Nintendo system. If you're getting one for the both of you, I'd go PS2 as it has a few kiddy games (Jak and Daxter). Xbox is for a much more mature audience and people who like PC gameing but would rather have a console.