Another option is used - there are often good boxes, proven reliable, that are sold off when the owner upgrades.
This is what I was thinking, buy used components mostly but maybe 1 or 2 new parts, like a Corsair CX430 that often goes on sale with MIR for <$20 and a new HDD. You could probably build a Core 2 Duo e7000-series system for ~$125 with DDR2 (est $30 for CPU, $55 for mobo, $40 for 4GB RAM - sadly since DDR2 is not in high production it isn't cheap, you could knock half off and go with still-sufficient for the stated uses 2GB for ~$20 though) +$60-$70 for a good sized 7200 RPM hard drive
and then have room in the budget for peripherals and an OS.
A C2D like that would be faster than an A4 or A6 for the uses you stated, it would be slower only for the most heavily multithreaded tasks but not be needlessly expensive like a C2D e8000-series. Usable keyboards and mice are cheap (I got this mouse
for my parents for 12.75 with a coupon code recently) and might bring us up to $200-$225. (Heck if you want a cheap IBM membrane keyboard PM me, I have one I can send for shipping) Add on the OS and a cheap monitor - check out your local second-hand charity stores like Goodwill - and you've got a good system with a new OS for $300.