Well, a distro is a distro more or less. Underneath, they all have the Linux kernel and GNU userland. If the distro is installed such that it's not just text mode, then they all have X11 installed also. There are also choices between window managers, from really minimalistic to very full-featured.
If I were running Linux on a very old computer, I'd find a distro that let me just install the minimal system plus X and TWM as the window manager. That choice of installability is the key factor here, not what distro you actually install. TWM is very lightweight and yet still is graphical and can run any other GUI program if needed. TWM definitely looks like something out of a 1980s OS but if you're not afraid of using a window manager sans glitz and a bunch of GUI tools, it'll do just fine. If you need more GUI than TWM's Xterms and Xclock provide, then IceWM or FVWM would work. XFCE is much more heavily featured than IceWM or FVWM are and would probably still run okay. But I'd really only use that if I really needed to. KDE and GNOME would probably run also, but I can tell you that they'd be slow and you'd need a fair bit of RAM to run them very well. They are full-featured modern desktop environments and perform well on newer hardware.
And as far as singling out a particular distro, I really like Gentoo. However, it would be a bad choice on such an old computer as it would have to compile at least some of its own packages and this would take considerable time. There is a dsitributed compiler, but it won't always work on every package and the computer would have to compile the package. That can take hours for certain packages. Thus I'd recommend any Debian-based distro that you can choose how much you install. Regular Debian would work, so would the server version of Ubuntu and a whole raft of other Debian offshoots. SUSE would work fine as well, but it's a little heavier than other distros and has fewer CLI config tools and more ncurses/X based ones. I haven't had much experience with Slackware or its progeny, so I can't comment on it. Nor on any of the BSDs as they're UNIX as well. I'd just say look at install options and any one that has a decently-configurable minimal install mode will do you fine.