I absolutely love the Gamecube so it's really tough for me to narrow it down to just 12 but here goes.
1) Smash Bros Melee
2) Resident Evil 4
3) Metroid Prime
4) Zelda: Wind Waker
5) Super Mario Sunshine
7) F Zero GX
8) Eternal Darkness
9) Soul Calibur 2 (the best version!)
10) Timesplitters 2
11) Luigis Mansion
12) Mario Kart: Double Dash
Honourable Mentions (if it were to be a top 20):
Prince of Persia: SOT, Mario Kart DD, Viewtiful Joe, Super Monkey Balls, Rogue Leader, Bloody Roar Primal Fury, Mario Party 4 & Sonic Mega Collection.
Twilight Princess would make the list except I own it on Wii not GC so it should be on another list. Also Paper Mario TTYD might do as well - I've heard great things yet sadly not played :/ it's really expensive in the UK!