You can fine tune nearly any Linux to run more lightly if you want. You might be best served picking a distro you already have some familiarity with and stripping away the things you can live without. If you don't want to put that sort of effort into it, then I would recommend Crunchbang. It's Ubunutu based & gives access to an impressive software repository, but uses a lightweight highly configurable window manager - Openbox. You can to 'roll your own' by installing a standard Ubuntu & then installing and cofiguring Openbox yourself, but the Crunchbang folks have done alot of the homework for you. They also include quite a few lightweight alternatives to the standard music / mail applications typically found in Linux distros.
It's almost as lightweight as Puppy linux running the e17 window manager, but has IMHO much better software selection.
One caveat: if you don't like the command line, then forget everything I've written here & pick something else ;-)
As The Journeyman Geek suggested, try out the Live CD's first & see which you like.
As an aside, I have Ubuntu 9.04 installed on a cheap e-Machines laptop I bought new for US$220. It's about as low end as you can get - she is very rough on hardware. My daughter uses it to check email, surf the web (facebook mostly) & listen to music, local and streaming. Believe me, if it wasn't working, I'd hear about it!