I believe a GT 240, GT 430, or a GT 520 from nVidia (these are all about the same performance, just different generations) would be able to run in your system without having to upgrade the power supply. They are low-end cards, but still more powerful than the integrated graphics. For an AMD card you can go all the way up to one of the new Radeon HD 6670 graphics cards for about $100 and it should run without needing a power supply upgrade. However, I wouldn't suggest anything more than that, and even a full blown dedicated card like the 6670 or GT 440 might be more than what the PSU has headroom for.
Keep in mind that a power supply will start to lose some of its capacity to output its total wattage as it ages. This means that a cheaper 300 Watt PSU (such as those found in the majority of consumer HP desktops) will actually only put out about 250 Watts or perhaps even less after a few years. So, depending upon the age of your computer and what all hardware it will be running now and into the future, it may be able to handle running the dedicated graphics card but then one day just start randomly shutting down because the PSU is failing. I have a PSU to replace in a Dell tomorrow because of this very issue after the customer decided to add in a second CD drive, hard drive, and TV Tuner card and the system is 2 years old. The PSU (this one is also a 300 Watt unit) just gave up and died.
The main thing to look for is selecting a card that does not require an additional power connecter (like a 6-pin or dual 6-pin PCI-E power connector.)