Super Mario Sunshine is in my opinion better than Super Mario 64, with a better atmosphere and setting. Occasionally the objectives can be a bit annoying but overall it comes together nicely.
Metroid Prime, one of the best FPS ever made. The second one, while not quite as good, is still great and continues the Prime plotline. The music and visuals create a world that few other games have reached the same level of, with memorable locations and battles.
Final Fantasy: Crystal Chronicles; Now, speaking about atmosphere, this game nails it. The controls are a little clunky, it has the unfortunate state of being used to push the Gameboy Advance link cables, and the mechanics are unorthodox, but it the game itself is fairly solid and captures the feeling of the world quite spectacularly.
Super Monkey Ball 1 & 2 are occasionally throw-controller-through-wall frustrating but compelling puzzle(?) games that also include a large amount of multiplayer minigames.
Kirby Air Ride tried out having extremely simple controls for a racing game, and it worked out decently. Three game modes: A fast third-person battle racing through creative and eccentric tracks, an overhead Micro Machines type thing with chaotic mayhem, and a RPG romp in an open city where you level up stats by collecting powerups and can switch between cars as you like. The City Trial mode and trying to complete the extensive checklist of missions are the key features.
Baldur's Gate: Dark Alliance; I don't care what some people say, this is a great game. An action-RPG hack n' slash in the style of the old Diablo games but with a greater focus on story. Has co-op same screen multiplayer and difficulty modes that scale up into the ridiculously hard.
Godzilla: Destroy All Monsters Melee is a little rough, but if you like kaiju and the ability to pick up buildings and beat your friends to death with them it's definitely worth looking into.
Soul Calibur II has likely the best singleplayer of any fighting game with it's Weapon Master mode, but is almost impossible to find a copy of.
Chibi-Robo was an underlooked gem, where you play a tiny household robot in a big house full of Toy Story style characters who occupy the rooms at night. Cleaning and doing good deeds gives you experience which you can spend on upgrades, and the power system is a clever way of managing the player by requiring you to recharge your batteries at sockets located around the house.
Games I don't need to make a writeup of, but are also fantastic are;
The Legend of Zelda: Twilight Princess and The Wind Waker
F-Zero GX
Beyond Good and Evil
Killer7
Super Smash Bros. Melee
Tales of Symphonia
Animal Crossing
Paper Mario: The Thousand Year Door
Fire Emblem: Path of Radiance
Pikmin 1 & 2
Luigi's Mansion
Resident Evil 4 and Resident Evil 1 Remake
Metal Gear Solid: The Twin Snakes
Ikaruga
Sonic Adventure 2 Battle
Prince of Persia: The Sands of Time
Eternal Darkness: Sanity's Requiem
Viewtiful Joe
Baten Kaitos Origins
X-Men Legends
Games you might also be interested in;
Sphinx and the Cursed Mummy
Alien Hominid
The GameCube had a variety of unique games, it's a shame a lot of them aren't remembered anymore.