My favorite games by system:

NES: The Legend of Zelda
SNES: Final Fantasy III (VI)
Sega Genesis: Pirates' Gold
Game Boy: Final Fantasy Legend II
Game Boy Color: Pokemon Yellow
Game Boy Advance: Final Fantasy Tactics Advance
Nintendo 64: Goldeneye 64
Playstation: Final Fantasy VII
Playstation 2: Kingdom Hearts 2
Playstation 3: Mass Effect 2
Nintendo DS: Radiant Historia
PSP: Tactics Ogre: Let Us Cling Together
Gamecube: Super Smash Bros. Melee
Wii: Super Mario Galaxy 2
PC: Final Fantasy XI

I thought that would be hard but it was actually pretty easy. Except PS3. That system is too recent to have a definite favorite.