A few of my favourite games are for Gamecube! Paper Mario 2, Animal Crossing, Zelda: Wind Waker, Smash Bros. Melee, Mario Kart, there's loads!
Paper Mario 2 is easily the best Paper Mario, Mario RPG, and Mario overall. It was the last Paper Mario game before the series got ruined by Super Paper Mario and Paper Mario Sicker Star (I can talk, Sticker Star isn't out here yet). Great story, great characters, great sidequests, and tons to collect and do. It's PERFECT. It's my favourite game of ALL.
Animal Crossing is a series where you just relax and do whatever you want. There's no fighting and is Perfect for casual gamers. There is no story, no main goal, but what you typically try to do is catch bugs, fish and dig up fossils to sell and upgrade your house or to donate to the museum. Huge variety of characters. You can buy clothes and furnature and all sorts. It also revolves around real time and relies on the Gamecube clock.
Zelda Wind Waker is typical Zelda game. As always, variety of characters and a huge world to explore, except this time you're on a boat and the world is 1 big map. Plenty of story, sidequests, collectable items. Tons to do.
Super Smash Bros. Melee is a typical Smash Bros. game. It's the hardest of the series.
There are other great games for Gamecube, such as Mario Kart Double Dash and Super Mario Sunshine, though those are 'series oddballs'. I find the best Gamecube games are those published by Nintendo.