Best performance depends on the game you're playing. The GF ti4200 and the ATI 8500s are very similar in speed on average - some are good at some games, others are good at others. However, the ATI cards look better: they are crisper in 2D performance and the textures in 3D games have always looked better to me.
I used to be a GF fan, but I wouldn't recommend them over ATI at this point unless you are going to be using the card with a specific game that is known to be run exceptionally well with that particular card. If it's going to be a non-gaming machine, the ATI cards have a better 2D display. The Radeon 8500s were so "sweet" in a general sort of manner that ATI stopped producing them and subsequently released three inferior cards that seemed better because of the higher numbers on them: the 9000, 9100, and 9200.
That said, if you would consider spending a weee bit more ($70 or so), I would recommend an Radeon 8500 (non-LE). Next choice is a Radeon 8500LE or a 9100 (same card). Note that with these cards, the 64MB models tend to perform better than the 128MB models (unless you really need all that extra vid RAM - which I doubt you will considering the stated purpose of the machine).
Stay far away from the FX vid cards. If you really want to get a cheap card that performs relatively well under light cirtcumstances, get an old Radeon 7000 or a GF2 MX400. You shuold be able to find either for $25 or so. I'd go for the Radeon 8500 myself. Much more bang for the buck.