No way I could pick just one. Gamecube has had some of my favorite titles to date. And it's impossible to say, because it's lifetime isn't quite over.
I have to say, Metroid Prime (both games), Wind Waker, Pikmin 2, Jungle Beat, and Paper Mario 2 are probably my favorite single player games. Pikmin 2 also had really fun two player, my friend and I laughed so hard.
Multiplayer goes to Super Smash Brothers Melee and Mario Kart: Double Dash, easily. Put tons of hours into these.
Four Swords (which I ever only played alone), Luigi's Mansion, and F-Zero GX get honorable mentions, for being quite fun but not as substantial. Well, F-Zero is pretty substantial, just not among my top favorites.
Killer 7 may also be among my top favorites, for being the first video game story I ever truly enjoyed (aside from Link's Awakening, which is my favorite game ever).
My all-time defining game for Gamecube, however, is SSBM. Played that game more than any other game ever, even more than Starfox 64, probably. So many fun times had with this game, it'd the defining game of this generation, for me.