Let's see....
Super Mario Sunshine was okay. If you aren't a big fan of collectathons you might avoid it.
metroid prime 1 and 2 are great FPS-vanias (I guess you'd call them that?), I still go back and play them periodically.
Paper Mario: The thousand year door is a pretty great RPG.
The first Viewtiful joe I loved, but in my opinion Viewtiful joe 2 is a pretty big downgrade. It's still a good game, so I'd just get 1 and if you like it move on to 2 maybe.
Pikmin 1 and 2 are great.
Tales of Symphonia is on the gamecube if you're a fan of fairly long JRPGs.
Some lesser known stuff...
Baten Kaitos is perhaps the longest game I've played and is also JRPG-ey, although the plot is generally a bit better then one.
Once more, Skies of Arcadia: Legends is a GREAT JRPG that if you enjoy the genre at all you owe yourself to play.
Chibi robo is a fairly bizzare japanese game where you are a tiny robot cleaning up a house, with a plot that involves much larger robots, energy efficiency, spider robots/aliens I forget which, and normal aliens. I loved the hell out of it but I do love quirky junk. If you want something different it's probably good to watch some videos first and see if it sounds like your bag. I wouldn't buy it blind.
In a similar vein, Cubivore is a game where the entire ecology of a planet is cubes with little rectangles for various bodyparts. You run around and eat other cube animals and turn into different beasties depending on the number and color of the various rectangle meat things you eat.
Again, quirky, good, I still wouldn't buy it blind.
Killer 7 is a David Lynch film. More specifically it's an FPS-on-rails with puzzles that would make resident evil blush. You fight invisible exploding men with no eyes, enormous smiles, and an aversion to ever stop giggling unsettlingly by shooting them in glowing weak points which you have to scan to find. You also turn into multiple men via way of a room with a TV and a transparent gimp hanging from the ceiling via a little wire attached to his harness inside of it. The plot is completely incomprehensible but still a lot of fun to see happen. You also fight the power rangers in a gunfight. I know it sounds like I'm making fun of it but all that happened and I loved the hell out of it. But I also like David Lynch! If you like wierd surreal shit overlaid over a fairly typical "assassins kill a bunch of people" plot you should probably look into it. I wouldn't look at any plot before you do which is unfortunately most of the cutscenes. If you can find a gameplay video then that should be fine. But the gameplay isn't a super strong suit (although it does keep things fresh, you'll generally be playing just because the plot is so bizzare).
I think that's about it! Gamecube was kind've a bad time for nintendo but there's still quite a few gems kicking around in there.
EDIT: itisnotlogical - There's one fire emblem game on gamecube I think, and it's.... pretty normal fire emblem. Ike is easily the most lethal lord in the game to the point many people suspect a solo run with ike would be easier then a normal no death run past the first few missions. It seemed a little bland to me honestly. The next one is on Wii and seems much better to me, but that may be just because I've not played it.
EDIT2: Oh, I'll also vouch for Sonic Adventure 1 and 2. A lot of people like making them look TERRIBLE OH MY GOD but they're honestly not too bad. The little chao garden feature in SA2 I probably spent way more time then I should have with.