That's tough. The First Xbox introduced online play which was unheard of for the tv. Ridiculously fond memories with friends for Halo 2 so that alone carries a lot of weight.
The 360 introduced one of my favorite franchises, gears of war which also stacks heavily in my mind.
I'd have to edge it to the 360 even though there were more games that I felt I enjoyed with the first Xbox (psychonauts, republic commando, battlefront 2, halo 2, the first Fable)
Dreamcast had a good following online. Phantasy Star Online, NFL 2k and Nba 2K.
360 may be the best console ever. The OG Xbox holds a ton of memories for me as well.