lets see.

NES: Nintendo System Cereal
SNES: Owned a NES

Sega Genesis: Sonic the Hedgehog
Game Gear: Prince of Persia (it had color!!)

N64: Legend of Zelda: Ocarina of Time (I even had a fan site and had 100s of pics)
Game Cube: Brandname

PS2: Burnout 2/DDR Max

Wii: Wii Sports (almost didn't buy any system at all)
DS: Legend of Zelda: Phantom Hourglass

well for gc it was RE4
for ps2 i bought it for its huge library
and for wii the ninty games mostly wii sports and TP

Dreamcast: NBA 2K. This game bascially SOLD me on the Dreamcast. Played it by a friend one weekend and couldn't shake the urge to own the game. I basically went an entire year on Dreamcast with that being the ONLY game I owned. I HATED NFL 2K and preferred Madden, then something strange happened: I dropped Madden for 2 years and ONLY played NFL 2K1 & 2K2!

Hmmm ... I see that many people really went for high-quality games only , still there are a couple of exceptions :)

This is going to be a same-y topic, because most people buy a system for the really popular games. There might be games you end up liking more in the long run or something, but something really big usually makes people pick up a system (Halo3/Gears, Metal Gear, Zelda/Mario, Sonic 1/2)

NES: super mario bros

game boy: I honestly can't remember

SNES: super mario world

n64: zelda OoT

game cube: smash bros melee

wii: in complete honesty - wii sports

ds: impulse buy - got new super mario with it though

360: anticipation for fable 2 mixed with tired of not being able to find a wii

if i ever get a ps3 - blu ray is a standard (not just beat HD DVD but also gaining significantly on DVD), ps3 is no more than 100 dollars more than stand alone players at that time, and the price of blu ray discs are

Either Mario or Smash Bros for consoles every time except GTA for PS2.

Pokemon for all handhelds.

Wii - Twilight Princess
360 - Eternal Sonet & Mercinaries 2
* I haven't bought the 360 yet, but when I get an HDTV and the price drops to around $200, these games are why I will buy it.

PS3 - nothing yet, but it is still early in the console life.

Since I have owned too many systems over the years, I'll just do current generation:

360: Gears, promise of 2007 Halo
PS3: There wasn't really one but Uncharted was what I looked forward to most
Wii: Twilight Princess (later sold it), and once MK, SSBB, and SMG are all on shelves, I'll buy another