My "roll your own" wireless access point was set up on an old AMD K6 that my dad gave me. Any old machine
from a P100 on up can serve as a simple access point, but set-up and maintenance are much easier if your machine
can run a fairly recent Linux distribution. Also, if you plan on running any sort of server, more up to date
hardware is definitely a big help. My access point currently runs Apache, IceCast, MySQL and a few other
things (try THAT with your puny Linksys, haha) without too much trouble, although the traffic it serves is very light.
As far as which distro to use, I originally chose Red Hat 9 because it had modest system requirements and was fairly easy to set up even to me, a Linux newbie at the time. In fact, there are pre-rolled Red Hat kernels available that have HostAP support already added in. They are 2.4 kernels, though, so you may be better off downloading the latest HostAP driver, compiling it, and installing it yourself more current. Distros built on 2.6 kernels, such asSuSE 9 and later, already include HostAP, wireless tools, and of course a slew of other useful things. FreeBSD 5.2.1 also comes with it. I switched from RH9 to Suse 9.1 and except for a few minor configuration issues, HostAP installed and ran great, almost "out of the box."
But if, like me, you need some help getting started, have a look at some of the   very   helpful   tutorials   that are available. These were invaluable for fixing many of the problems that cropped up.