Many, many games from that era were just awesome, but the ones I truly enjoyed when my Cube was still working and kicking and when I still had my PS2 before 2004 were:
Cube:
*Wind Waker
*Eternal Darkness
*F-Zero GX (much, MUCH better than F-Zero X, in fact, this may be my favorite racing game of all time)
*Soul Calibur 2
*Paper Mario: Thousand Year Door (a Thousand times better than the N64 original)
*Metroid Prime (you must did something right if I still consider Prime one of the best games for THIS generation with Trilogy)
*Super Mario Sunshine
*Resident Evil Remake (I don't know why many people don't like this game, this game scared the shit out of me more than the original)
*Resident Evil 4 (the Wii version is better, but I still enjoyed this game a lot)
PS2:
*Metal Gear Solid 2: Sons of Liberty (my only reason to own a freaking PS2 and the wait was so worth it, with flaws and all)
*Devil May Cry (love at first sight)
*Onimusha 2 (the first game was great and the sequel was better, I wonder why Capcom hasn't give some love for good ol' Onimusha yet)
*Silent Hill 2 & 3 (still some of the best horror games I've ever played)
*Kingdom Hearts (before Square made the story bat-shit crazy, before the spinoffs, there was only a Keyblade and lots of baddies)
*Final Fantasy X (I never actually finished this game, but up before facing Jet, it was a wonderful journey up to that point)
*Metal Gear Solid 3 (I played this game with my cousin after I stupidly sold my PS2 and this game made me regret it so much)
*God of War 2 (same with MGS3, awesome game)