Yeah, what Rogue Leader said. The 750 Ti or 950 from Nvidia, or the 270 series (inc. 270x) or 370 from AMD. For 1080p, all of these cards are good. The 750Ti is a particular standout as it only has a 65w TDP and can work with some (strong) 300w PSU's. I wouldn't try it with an off the shelf PSU that's in computer that old though. PSU's degrade over time and get worse as they get older and eventually fail.
Even for a 750Ti I'd still grab at least a 450 or 500w PSU though. From someone reputable, like Antec, Corsair, or Seasonic or someone lesser known that has the same quality, like XFX or Cooler Master. Getting a 500w PSU (or thereabouts) gives you some upgrade headroom too.
Personally, if price is a concern, I'd grab an old 270 or 270x. A 270 in particular can be overclocked to the speeds of a 270x, as it's the same card, but with lower clock speeds and power usage. The 750Ti is also the weakest of the cards listed above, so there's that to consider as well.