I would say 512mb just to save you from any possible frustration. My parents computer is my old one: Tbird 1.2ghz, 512mb RAM. That thing pretty much screams for them. I formatted it a couple years ago when I got my new system and JUST by using Firefox it has not gotten any spyware or any crap.
My work laptop is a P3 1.2ghz and 128mb ram. It can get pretty slow but I was doing MPLAB with ICE emulation, several browser windows, and misc. other small-ish programs without huge slowdown.
I just reformmated XP on a friends mom's computer. Its a P3 1GHz with 128mb ram. Much to my surprise it ran decently. I made sure to turn off anything unncessary as well as setting visual to fastest performance.
Again, with the 128mb systems its really pushing it but if you strip it down of unneeded running programs (like system restore for one) its still do-able for basic web browsing, email, and office apps. On these systems its not worth it to have on-demand spyware, virus, firewall, etc protection (well even on a 2GB system I wouldnt do that crap) when you can just make a restricted account and save the RAM for using the damn thing.
So theres 2 128mb systems that I can still use and not get pissed off with for doing basic things. But from what I've used with my parents computer, 512mb is your best bet for staying barebones but speedy.