1) I would try the latest Slackware first with fluxbox or blackbox Window Manager and use the Dillo web browser.

2) Vector Linux with Dillo http://vectorlinux.com/

3) AmigoLinux - which might be your best bet if the machine doesn't have a bootable cd-rom. This is a minimal Slackware installation that you simply unzip to a DOS partition. You can use something like the Tomsrtbt floppy linux to use wget to download the zip file to the partition http://amigolinux.org/