Yep, if you only build websites but not hosting them live, any old computer can do the job pretty well; however, if you just like to have a new computer, build or buy a cheapest one for around $150 or less will be better than enough for designing websites and even hosting them live.
Recently, my friend ask to help him buying a cheapest computer I can find, and finally he's got SR5010NX for about $120; then I've helped to setup wamp, ftp server, mail server, so on for him to have a live forums, blogs, so on, and so far still so good
Have fun,
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