Your processor is fine. I would like to see the definition of a "weak processor". Your memory is adequate at 1 GB. Most power supplies are No Name, and the name means nothing except the seller has money to advertise and market. There are 87 different power supplies that I know of, with "no name," and some such as FSP and Sparkle are sometimes incredible. For sure, you need adequate voltage, and steady delivery under load which can instantly respond to the demands of the video graphics card. You can read excellent reviews on thoroughly tested power supplies at various sites,
www.tomshardware.com for one. If you have a stock system such as an HP, you will definitely need a better power supply to handle a good AGP video card above 32 MB.
I second SNGX1275's suggestion of the 7600GT for your configuration.
It appears you have a Foxlighter motherboard, or perhaps an HP machine. The motherboard came out in the fall of 2005, but has been trouble free.
On AGP video cards, you will find them at 512 mb, but you will detect no difference between a 512 and a 256.
If money is an issue, the Sapphire X1650 Pro or X1560 XT can be found for $93 in either the 256 or the 512.
But look for the tested, reviewed performance... which is why the 7600GT is a fine choice, assuming your power supply is decent.
You can also use the power supply calculators such as the
http://www.extreme.outervision.com/psucalculator.jsp for some helpful info.
There is a great deal of helpful info on this forum about power supplies.