I only started buying my own systems in the middle of last generation (already had a PS2/Gamecube, bought my own Xbox), but I'll say what I wanted them for.
N64: Mario 64 (I'd actually never heard of the N64 or Playstation till my bro told me about the N64 a few days before he picked it up )
Playstation: (Don't laugh) Small Soldiers. I LOVED that movie
GBA: I just wanted a portable SNES.
Gamecube: Luigi's Mansion, Super Smash Bros. Melee
PS2: My bro and I got this for a X-mas present the year it came out--didn't really know the games
Xbox: Star Wars: The Trials of Obi-Wan, May Payne (thought Halo looked stupid until I tried it XD )
DS: Mario 64
PSP: Twisted Metal: Head On, and I'll soon have to rebuy one for Kingdom Hearts.
Xbox 360: Dead Rising, Perfect Dark Zero, Condemned: Criminal Origins
Playstation 3: ...nothing, really... got it as a present... wasn't that interested in it at the time, to be honest... was looking forward to Lair but... yeah...
Wii: The controller, Wii Sports, Zelda, Red Steel (hey, it looked good before launch, k?), Mario Galaxy, Brawl.