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If you're going to use dreamweaver or something, you'll need to check their site for minimum requirements. Same with the ftp client, graphics editor, browser to view the site, etc.
I imagine crimson editor, filezilla, firefox, and the GIMP will probably run on some old hardware. And that's really all you need to build a site.
I'm starting to experience hardware problems, but if I wasn't I would have no compelling reason for needing to upgrade.
You can build websites with any old computer, as long as you can run the latest browsers. I think the minimum recommended for Firefox is a 233Mhz and 72Mb of RAM. If you're running Windows, IE's requirements are even lower (however other Windows programs may need more power).
There are so many more important things than computer hardware that you can spend your hard-earned cash on. Especially if you're short of funds, keep your laptop and invest your money elsewhere.