I have heard others refer to CDLinux (find it here: http://sourceforge.net/projects/cd-linux/ ) as a good one for extremely limited-resource machines like yours. I have not used it, but here it is as a suggestion to check out.
Another is Portable Linux, a SliTaz fork, that sounds interesting -- http://portablelinux.sourceforge.net/
Seconding Goineasy9, I have antiX installed on an old PIII with 512MB RAM and hear it runs swell on a PII with 256MB RAM, but this is probably the minimum regarding system requirements. The developer for SaxenOS also used this latter as a description of the minimum system for his distro in a webpage read recently. Both run very well on my PC.
I all you want to do is surf the web and save small files occasionally, the new Lucid Puppy with a USB stick or Slax with its free "Slax Drive" could serve you well running as LiveCDs.
http://www.linux.org has a list of distros, also, that may be helpful.
Best wishes in your endeavor!