I own a gamecube and I must say I have hardly ever played as the great games are pretty far and few between each other. Theres nothing wrong with the system. Its great, the games look awesome, they play just fine and keep me entertained for a hours at a time, even for games I bought a year ago.
The majority of people own PS2s, it has a huge library of games so theres a chance you can find a good game for everyone who wants one, but since it has so many games theres a nice percentage of those games that are more towards mediocre.
Xbox would be for the person more geared towards "mature" games, and who wants action titles and high quality graphics. Most of the games they make are being made for going online and playing multiplayer.
So the older kids may not care for a gamecube and the the Xbox may not exactly be full of games the younger kids can really dig. By the process of elimination the PS2 seems to be the system of choice. Just don't buy the little kiddies the 18+ year old games. Can't stand listening to a horde of 10 year olds talking about killing people in grand theif auto games and saying its the coolest thing ever.