I decided to look at each seperate year of the Gamecube and pick my favourite game for each year it was alive. I'm starting in 2001 and ending in 2005
2001
Pikmin

Runners up: Super Smash Bros. Melee, Luigi's Mansion
2002
Resident Evil

runners up: Metroid Prime, Super Mario Sunshine, Animal Crossing
2003
Beyond Good and Evil

Runners up: Zelda: The Wind waker, F-Zero GX, Viewtiful Joe
2004
Pikmin 2 (best game ever!)

Runners up: Metroid Prime 2 Echoes, Paper Mario The Thousand Year Door
2005
Donkey Kong Jungle Beat

Runners up: Resident Evil 4
enterawesome