MGS:Twin Snakes is a faithful remake of the original MGS, and the surreal, pure platformer levels where Mario loses the FLUDD make Mario Sunshine worth a try.
Tales of Symphonia was the only JRPG I ever got to play on the gamebox, and it hooked me into the genre.
I don't know of many horror games on the gamecube, but Resident Evil 4 immediately comes to mind and there are some gamecube ports of classic RE games I remember owning.
Whether you love the fantasy racing genre or not, pick F-Zero GX. The 'Very Hard' "Story Mode" (cheesy story of Captain Falcon saving the planet) presents interesting challenges once you've learned to play the game properly (i.e. abusing the game engine through "snaking", making your machine ridiculously fast), and you may appreciate the fast paced racing and the techno-rock music (F-Zero X players might miss the older rock OST). I think there's a Mario Kart game for the cube as well, "Double Dash"
The original Animal Crossing is a great 'stoner' game (I don't know what genre it would fall into) to kill time.
Don't get the cube and forget to get the classics like Smash Bros. Melee, Metroid Prime, Wind Waker, Pikmin, and Luigi's Mansion with it. All the above games made me love the gamecube more than the xbox and the PS2 despite its inferior hardware. I've missed tons of great games for the gamecube that I'm sure the more savvy members of the community will nevertheless suggest for you.