Since I don't know what you have played or what your tastes are, I'm just going to recommend some GameCube games which I enjoyed.
Super Smash Brothers Melee
It's different enough from SSBBrawl that I'd say it's worth checking out.
Luigi's Mansion
AKA the point when Luigi stopped being a neglected character. It's short but it is enjoyable while it lasts, the controls are solid, and the real challenge is going for a high score.
Super Mario Sunshine
I don't care what you might have heard about this, it's still a great game that's just a little rough around the edges.
Metroid Prime 1 and 2
The atmosphere that these games manage to create is simply incredible.
Wario World
This game might be short and simple but it is quite enthralling, and the stages are pretty varied with the small pure platforming segments tucked away under the ground panels. I suppose that structure is a bit like Mario Sunshine, but there’s nothing wrong with going with what works.
Battalion Wars
The game brings you a lot closer to the heat of battle than the average strategy game. And with direct control over one of your units, you really feel like you're causing some damage, rather than just watching a staged battle. Battalion Wars is a great mix of strategy and action, but you'll need to get past the first couple of missions before the real battles begin.
TimeSplitters 2
"Heir apparent to Goldeneye." Believe it.
TimeSplitters Future Perfect
It's a little rough around the edges feature-wise, but the gameplay is enhanced and it has more variety than before. The story mode is a bit short but it has some epic moments when Cortez meets up with his past and future self. There’s lots to do in the Arcade League and Challenge modes; going for high-scores is, well, challenging! The stage builder is a lot more robust and is capable of creating fairly complex arenas, I've spent many a sleepless night just building away.
Burnout 2
A pretty solid arcade racer that rewards dangerous driving by awarding you speed boosts for driving close to traffic, performing stunts and drifts, etc. The crash mode is also a ton of fun.
Paper Mario: The Thousand-Year Door
It can sometimes get a bit tedious due to its simplicity, but it is still an engaging RPG which will magically suck up all of your free time until you complete it.
Tales of Symphonia
The more action-oriented battle system kind of reminds me of Smash Brothers in the way that the controls are set up, and with how the movement is mostly two-dimensional. Although most battles aren’t really difficult, what makes you always try to play at your best is the grade you earn from performing well. The story is pretty clichéd and conventional, though the random skits which pop up every now and then are a nice touch.
Mario Golf: Toadstool Tour
This is a solid golf game, despite that the swing system is exactly the same as NES Open Tournament Golf for the, uh, NES. If you like both golf and Mario, then this game is worth owning. If you like one but not the other, then you may want to consider something else.
Mario Kart: Double Dash!!
Awesome title, that. How many game titles have not one, but two exclamation points? Not enough, that's for sure!
Mario Superstar Baseball
The best thing about this game is that it contains 32 characters, so you'll see a lot of playable faces not available in many other games. The controls are a bit imprecise and take some getting used to, but work fine once you do. Fielding can sometimes be troublesome though, as movement is a bit slippery and there's no way to switch control of fielders. Still, it's nothing you can't anticipate with practice. Though if you have the Wii sequel then you probably needn't bother.
Super Mario Strikers
This game isn’t “Mario” enough in that it doesn’t offer up enough twists on the gameplay and the characters don’t have very distinct personalities. Even so, the gameplay is solid, and the 4-on-4 makes for a fast action-packed football game. I prefer this over the extremely contrived Mario Strikers Charged on Wii.
Mario Party 4, 5, 6, 7
Everyone should own at least one Mario Party game... You never know when a party is going to break out! I suppose there's no reason for you to own more than one though, and I would recommend 7 as it has the most features.
Are there too many Mario games on the list? Don't be silly, of course not.