Legend of Zelda: Windwaker
If you've never played a Zelda game, this is NOT the one you want to start with. Sailing sucks. I can't bring myself to play this game again because of that reason. (I read long ago that sailing only accounts for 12.5% of game time, not including side quests/extras. Yet 1/8 of my time on a freaking boat, staring at a screen doing nothing is droll. Oh, and that 12.5% is if you know where to go exactly, and take the straightest path possible. The ocean is simply too large in that game.) I want to play it again, but just thinking about that boat...

Also, I thought it needed 1 or 2 more full dungeons. Majora's Mask on the 64 suffered the same pitfall, but made up some of that ground in complexity and size of the dungeons. (The flippable one was truly outstanding.)
On that note, try getting the original+remake of Ocarina, and the Zelda collection as suggested above. (Has Majora's Mask on it, along with the NES games. Does NOT have the SNES one though.)
Both Metroids...if you liked any Metroid before, you should LOVE these. Both are outstanding. What's really nice is that Echoes' Aether/Dark Aether truly felt like an entirely different planet from Tallon IV. Nothing felt recycled. Different controller scheme only takes 5-10 minutes to adapt to.
Smash Bros. If you like multiplayer beat-the-crap-out-of-each-other mayhem, this one tops them all. Insane fun
Mario Sunshine. Just a Mario 64 remake. Not nearly as fun, with one exception. The shadow Mario levels. Rest felt like a chore. Especially getting all the blue coins. Even when you had an FAQ to find them all. And one more thing. Watermelon=frustration beyond anything you could believe. How could something so seemingly simple on paper turn out to be so damned infuriating?

DO NOT BUY STAR FOX ADVENTURES!!!! My vote for WORST. GAMECUBE. GAME. EVER. First and only game I ever bought, then sold soon thereafter.
Didn't try Star Fox: Assault yet.
Mario Kart rocks. Another good 4 player game. It can be played online with the proper software and such I believe.
Which Mario Party? Probably 5. Haven't played 6, and while I like 7's idea of a departure from the normal party game, I thought they should've left the OPTION of a normal party game in there for all boards.