I'm just going to go ahead and list out from my collection of (awesome) Gamecube games (regardless of how many times they've been mentioned).
Metroid Prime is a classic, absolutely a must play.
Metroid Prime 2 does not quite reach the heights, but for the atmosphere alone it's worth it.
Legend of Zelda Wind Waker has always been a personal pick, I feel it's the best in the series.
Battalion Wars a quirky little real-time Advance Wars spin off, had a ton of fun playing it.
Super Mario Sunshine oft maligned, but wholly enjoyable
Chibi Robo a hidden gem I poured hours into this game has a wonderful charm to it
Pikmin 1 & 2 same as above, the second game is superior shines with Miyamoto's trademark sheen
Paper Mario (title too long) is the best in the Mario rpg series
Mario Kart Double Dash is a blast with friends
Super Smash Bros Melee same as above
Doshin the Giant again, not mentioned much, misunderstood I think, it's a wonderful ambient game
Wario Ware the gamecube version was the best
Animal Crossing if you haven't played the others, give this one a go
Resident Evil the remake is a great survival horror experience (4 is some good cheesy fun too)
F-Zero GX fun, challenging racer
I think I'll leave it there... too many games, feel free to p.m. me if you're looking for any more recommendations!