you might want to look at something at least 128bit

the pickings are slim
my last card was a pci 9250, it was pretty decent as far as pci cards go

 

Scorpei beat me too it, a 8500 GT is a good choice, it also runs some games fairly well, fairly as in low everything

 

The 8500 is (roughly 7600 performance), IMHO, worth the extra cash over the 8400 (roughly 7300 performance). Getting something for the same price on PCI-E or AGP would have been nice though . You could look for something second hand?

As for dolphin, not a clue, don't know if they use your graphics card a lot (doubt it) or if the CPU is the main factor (probably). If you really want I could try it on my 9300M G (core 2 duo T5750) which is the closest I have to such hardware, unless you are willing to wait till next wensday: my server should be built then which is composed out of:
AMD athlon 64 3200+ (newcastle, socket 754)
ATI HD2400 pro AGP
So the processor there isn't all that great and will probably be the limiting factor.
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Just checked for you on my laptop (9300M G). CPU use is pretty high ('80%-100%') but the vid card is used too. I get (with audio turned off, not working anyway with Zelda the windwaker) roughly 20-25 fps. Playable but not super. Audio is not working with Jungle beat either, fps and cpu use are roughly similar too.

I think you're better of buying a genuine gamecube (you can even emulate lots of things on there anyway) .