In reply to stp:
A computer is a multi purpose machine that does everything with various forms of success. a games machine does one thing, brilliantly!
The 360 is the benchmark IMO, the PS3 is great if you will use the bluray DVD player on it.
The PSP is a bit flash and costly for a 6 YO, the DSI is something you'll have to ask yourself 'is it worth the extra money for a crap built-in camera?' The bog standard Nintendo DS is superb value for money. There are some quite good educational games out there as well. My daughter is 5 and she's had one for 6 months, she loves the cooking mama and Dora/Diego games.
<if you are bittorrent savvy look at an R4 cartridge or similar. £20 for an R4 + 1 gig card, 25 games from the torrent sites and away you go. Not that i condone such things though>