Oh man, Gamecube, the lunchbox powerhouse. I assume you are gonna ROM these cause many of them would prolly be difficult to find on your own nowadays. Either way here's my lengthy list:
Tales of Symphonia (One of the best RPGs ever)
LoZ: Wind Waker
Final Fantasy Crystal Chronicles (more of a multiplayer game but still good single)
Harvest Moon A Wonderful Life (best game in the franchise, easy)
Animal Crossing (If you never Played one)
Custom Robo
Super Mario Sunshine
Luigi's Mansion (much much better than Dark Moon)
Billy Hatcher and the Giant Egg (an unsung SEGA game that deserves much more)
Chibi-Robo
Dance Dance Revolution Mario Mix (if possible, Waluigi is the villain yo!)
Dokapon Kingdom (Imagine Monopoly, Final Fantasy, and Mario Party rolled into one. Fun on single player but even easy mode is so god damn difficult)
Fire Emblem: Path of Radiance (Ike appears!)
Kirby Air Ride
LoZ: Twilight Princess (my favorite zelda game ever)
Mario Kart: Double Dash (best mario kart ever)
Metal Gear Solid: Twin Snakes (it's just solid 1 and 2 but if you never played them, do it)
Metroid Prime I + II
Paper Mario: The Thousand Year Door (the BEST Mario Rpg ever)
Pikmin I + II
Pokemon Colosseum
Pokemon XD: Gale of Darkness
Resident Evil 4
Skies of Arcadia Legends (a very good rpg game)
Sonic Adventure 2: Battle
Sonic Heroes
Super Monkey Ball series
Viewtiful Joe series
and finally, a game no one ever seems to know and one of my favorite games on the GC.
Cubivore: Survival of the Fittest
This game, is so goddamn fun, for only being a tiny box that eats other boxes in order to become the dominant box species in the world.