I use an old HP Pentium M laptop to take notes at school, would this machine benefit from the free space/RAM of Linux or are the benefits negligible? If so, which version should I use?
Here are the specs:
1.86GHz Pentium M
Some sort of ATI mobile graphics card
Generally (depending on your choice of distribution) Linux will use less resources than Windows. This can potentially result in a boost in speed and usability. Assuming you don't need any Windows-specific applications, I think it would be an excellent idea.
Anything with a light-weight desktop environment (LXDE, XFCE, Fluxbox, etc.) should run just fine. Here are a few suggestions of mine for distributions:
Debian w/ LXDE/XFCE