Most of the systems I've ever purchased were for more than one game reason, I think.
(this is only the systems I bought myself, I had previously owned a NES, Amiga 500, Mega Drive, Mega CD, Game Gear, game boy pocket + Colour, Sega Saturn and Dreamcast)
Gamecube: Before this point I mainly stuck to Sega consoles, so when Sega quit the system race I decided to hop over to a Nintendo system. I think it was probably Smash Bros Melee that sold me on it the most. That and Sonic at the time had made his home on the system, and I was hoping for a Sonic Adventure 3. Yeah, I know. :lol
In the case of the GBA I bought almost sorely for Wario ware after playing a mates copy. and getting hooked on it.
I actually caught up on the PSone sometime in 2001, mainly because I found one of those small ones for a fiver. I don't think any particular game sold me on it, it just had a lot to choose from. I mainly wanted to try out the oddities like Parappa and Vibribbon, though.
For the Playstation 2 it was actually all those retro arcade collections that came out in 2006 that made be finally buy one. Capcom Classics, Taito Legends, Midway Treasures etc. That and just like the PSone it has an insane amount of games for it, and buying this was like buying 2 systems and a DVD player in one.
I also bought a cheap X-Box at some point, mainly for it's Sega exclusives. Played JSRF and Panzer Dragoon on it and ended up selling it afterwards. Didn't particularly like it.
With the Playstation 3 it was most likely LittleBigPlanet though i think it was mainly it's features that sold me on it (60GB): to me it was all about buying something that was 3 consoles and a media player in one, and Sony consoles have been pretty faithful to me so far.