There's a lot I like about the GameCube. To this day, the controller is still one of my all-time favorites. I actually refuse to play Smash on WiiU without it. I have to say it, though, the D-Pad was absolutely awful. Thankfully, it wasn't needed very often. The system itself was neat, too. The purple color was pretty rad and I liked the small size with the handle, making it easy to carry to friend's houses to play Melee... not to mention the four controller ports. The GBA player was something else I really loved, as well as linking certain GBA games to GameCube games for secret unlocks, such as getting the Fusion Suit in Metroid Prime or unlocking the original NES version of Metroid - that was awesome.
When I got my GameCube I got it with Super Mario Sunshine and Metroid Prime. Those two games together had me in gaming bliss. A lot of people don't care much for Sunshine but I really enjoyed it. Soul Calibur was one of my favorite Dreamcast games so, naturally, I put an insane amount of time into SC2 on the GameCube, which, also had Link as a playable character and thus making it all the better.
The GameCube era was also my introduction to one of my favorite things in life: Smash Bros. Ah, the memories of when I first got into that game and the long, frustration road of getting good at it...
One genre the system was really lacking in was RPGs. Although, the few that were on it were fantastic. Twilight Princess (though I technically never played the GCN version since I got it at the launch of the Wii), Wind Waker, Baten Kaitos, Paper Mario, Fire Emblem: Path of Radiance... Shockingly, Skies of Arcadia and Phantasy Star Online saw ports to the 'Cube, though I never did pick them up since I already had the Dreamcast versions (perhaps now would be a good time to replay Skies, on a Nintendo system this time. Hmm)
Two of my favorite games ever got remakes on the GameCube, which are Metal Gear Solid and Resident Evil. While I didn't care much for the remixed music of MGS: Twin Snakes, the better graphics and new gameplay mechanics made it worthwhile. And that RE remake was nothing short of a masterpiece.
Another advantage the console had was that, due to a deal between Nintendo and Capcom, Resident Evil 4 and Zero were exclusive to the GameCube at launch and for a long time after. PS2 didn't get ports of those games for years, IIRC. And with RE4 arguably being the best installment in the series, that was huge at the time.
Something else I was super excited about during the GameCube's heyday was that we were getting home-console Pokemon games for the system. I was crazy hype for Colosseum before it came out. And while I did enjoy the game, it was Pokemon XD: Gale of Darkness that I truly loved. Something about that game really got me. The environments, characters, music... I thought it was great, though it's not highly regarded as one of the better Pokemon games. I sure put a lot of time into it and it's still to this day one of my favorites in the franchise. On a side note, I seriously can't wait for the first Pokemon game to come to Switch!
Other honorable mentions:
Beyond Good & Evil
Wave Race: Blue Storm