That HP comes with a 300w PSU so the Radeon HD 5770/6770 are not recommend since they have a maximum power consumption of 112w. The motherboard also seems to have the older PCI-e 1.0 or 1.1 slot. That means there is a small chance that it may not be compatible with PCI-e 2.1 cards; there have been people who reported such problems. It is unknown if a PCI-e 3.0 card will be guaranteed to work in such an old PCI-e slot.
This basically means you should avoid the Radeon HD 7000 series (PCI-e 3.0 card), unless you don't mind taking the risk of having to return the card if it does not work. As stated it is unknown if it will work in a PCI-e 1.0/1.1 slot. Most people in this forum seems to have motherboards with PCI-e 2.0 slots. The following is a Radeon HD 7750 for $110 if you want to take the chance; it is the most powerful graphics card that will work with a 300w PSU.
I recommend giving the Radeon HD 6670 a shot. There is a small chance it will not work in a PCI-e 1.0 / 1.1, but it is the best performing graphics card you can buy for a 300w PSU with a high probability that it will work in an older PCI-e slot. The following Radeon HD 6670 sells for $60 after rebate.
If you don't want to take any chances of potentially having to ship back a card that does not work, then you will need to buy a PCI-e 2.0 card which is 100% compatible with the old PCI-e slot. The problem is you will need to get an nVidia graphics card. nVidia do not make bad cards at all, but they tend to be more power hungry so only the lower end models will work with a 300w PSU. Having said that, the GT 430 is the best nVidia video card that I would recommend. You can buy the following one for $40 after rebate.
Below is a performance chart which uses the Radeon HD 6670 as a baseline. The nvidia GT 430 has 61% the performance of the Radeon HD 6670; therefore it is much slower. The new Radeon HD 7750 is not part of the chart, but it's performance is pretty close to the Radeon HD 5770 and GTX 550; so it's almost 50% more powerful than the Radeon HD 6670.