RAM memory failing? This is very rare... how do you know? it's the RAM failing? Not trying to be a wiseguy but maybe there's nothing wrong with the PC...
I would agree with the folks here who recommends the cheapest XP machine available with the following features.
- 22" minimum LCD Display DVI/HDMI driven (even the cheap ones look good these days), preferably with built in speakers and additional USB ports(clutter control)
- Dual Core Processor (any speed will be fine for web surfing + light photo work
- Dual layer DVD burner
- Minimum 2GB RAM, 500GB+ Hard Drive
- Integrated SDHC reader
- Load Avast antivirus, Openoffice.org and Google Picasa and you've pretty much got everything you need
Now... if you want to tinker a bit...
Get a hold of a Windows XP OEM CD (only the CD is required as you'll be using the XP COA from the PC)... blow away the factory install and do a clean, non junkware infested install of XP. Back up the factory install to a CD first though (there's usually an option to do this or in rare cases, recovery CDs). You might be surprised how fast these basic PCs are without all the bloat/trial/junkware that gets bundled from the factory. I personally prefer a very "lean" install without any background processes other than the basics.
My primary PC is 3 years old, single core Athlon XP, 2GB of memory and it's plenty fast (and I shoot RAW with my digital SLR).
For your old PC... The PC I am typing this on is my 2nd PC and is more than 6 years old. It runs Ubuntu Linux. I use it primarily for web surfing. Works great! You might want to burn an Ubuntu CD (free) and give it a go.