Don't look for a distro or a subdistro, look instead for tools to use to do what you want to do, and then check if you can run all these in that computer. You should detail what are the "basic needs" you will focus on. E.g., if you want to focus on WYSIWYG "office suite" document editing, then you're advised to try something lighter than LibreOffice (although if you don't open another memory-hungry application, I think you will manage to run libo).
If you want web browsing, you should see if you can get a lightweight browser. If you don't rely too much on dynamic AJAX-bloated sites and you really surf HTML, there are some choices out there, ranging from
(But Firefox should be able to run too — but unless they managed to get rid of the leaks, you have to restart it once in a while, also be sure to disable any overkill eye-candy and to disable scripts and plugins unless they're really needed (NoScript and Adblock or the like will be handy here).)
Your main bottleneck is, definitely, RAM. Try to get some more RAM if you want to run something heavier. Also, try using a fast hard drive for the swap partition.
CPU speed won't mean a lot unless you're going to do intensive computation, compilation, image processing, etc., or unless you choose a source-based distro (and, curiously, the bottleneck with source-based distros will be RAM, not CPU).
The choice of distro is meaningless as soon as you pick the tools you want to use. Of course, if some distro forces you to go through a default installation of KDE running firefox with thousand background processes to provide "automagic" behavior, then you will have some trouble setting it up.
I guess the best choice is, whatever distro you pick, check whether you have to use some special option to pick the packages and install the most minimalist system you can conceive, then install pieces as you find out you need them.
(A good generic tip would be: stay away from DEs (Desktop Environments) -- a window manager suffices to get, well, window management. But I guess whether you can get rid of some of the things DEs give you depends on how you work and on your tastes.)