I have so many Gamecube moments, too many to mention, really. But I'll name a few of the most poignant ones

Two on the first day I played it: Just laying my hands on the controller was amazing. It's by far the most comfortable and ergonomic controller I've ever experienced, surpassing Dual Shock, Xbox, SNES, N64 (blol), all of those. I just loved the way it felt.
And then marvelling at the brilliant graphics in Luigi's Mansion. Which are still some of the nicest looking ones this generation.
Super Smash Brothers: The best party game ever, period. It was the only game I had to play for six months (after beating Luigi's Mansion) and I played it nearly every day, and I still play it fairly regularly, especially with friends. It's just awesome to play with friends, smack talking and trying different modes (Bob-omb matches) and generally having fun.
Also, playing it with my friend murk. He and I are both incredibly good at it, so it's tons of fun to play it with him.
Metroid Prime 1 & 2: Both amazing games, new and fresh. I freakin' love them. They really got me into the 2D Metroid games, too, so I played Super Metroid, Metroid Fusion, and Zero Mission.
The first is particularly memorable. Like MJrx, I played it during Thanksgiving break, and my family always vacations in Florida. Playing the game, aside from being great in its own right, reminds me of playing it there and what a great trip that was.
No particular memories tied with 2, but I really liked it anyway. Beat it all over on hard mode (plus 100% on normal, items and scans), and enjoyed every minute of it.
Except boost guardian.
Wind Waker: The graphics just blew me away. I still play the game sometimes just to look at them in awe. And sometimes I still play it for the fun part, too. I've completed virtually every sidequest and I'm still finding new stuff. I found a new enemy cave on Pawprint Island just a few days ago. That's a wonderful feeling.
My second favorite Zelda game for a whole slew of reasons.
F-Zero GX is great for being a ridiculously hard game. Never cussed so much ever, except for in SSBM.
Pikmin 2: One of my favorite games, period. The first day I popped this in, I had no class for several more hours. It was raining outside, and my clothes were wet. I changed into dry pajamas and sat on my floor about five hours, just playing it. It was just like being a kid again, waking up early in the morning and playing before everyone else woke up. Incredible feeling, I loved it.
Super Mario Sunshine: Playing this game makes me feel like I'm on vacation. I never bought it (borrowed it twice, extensively), but I'm still tempted to do so just because I simply love playing around in it.
So many others, too. Heck, I might list more later just for the hell of it.