Find out what his PSU is; brand and model, not just [claimed] wattage.
If he's running a HD5770 now, he should be able to run a HD7850, which doesn't use much more power (still only needs one PCIe power connector). That should give him the performance he wants, and leave him money for a PSU if he needs it. In fact, if he maxes his $300 budget, he might be able to get a HD7870 (XFX DD for $230) and a PSU (Seasonic modular 520W for $70).
Thanks! I will report back when i know the wattage/brand/model.