Most any current distribution will work fine on this, even though the CPU is a bit pokey. Try Linux Mint (derived from Ubuntu, which is in turn derived from Debian). If you want a Red Hat Enterprise Linux (RHEL) system, you can install either CentOS or Scientific Linux (SL), which are free clones of RHEL. I use RHEL 5.4 and 6.1 at work, SL 6.0 and 6.1 at home, and used to use CentOS 5.x, but switched to SL a bit over a year ago because CentOS was slow in adopting RHEL version 6, which has major improvements over 5.x in hardware support, and current software such as audio/video editing tools.
If a Windows can run on your PC then linux will 100% run on it and twice as fast. Linux uses minimal resources, not like windows ... Try Ubuntu to get started and then move to Debian or if (in present) you had to install linux on a slow pc then install debian :D
What type of graphics processor does it have? I've had trouble in the past getting certain distros installed with certain NVIDIA graphics sets. If you are having a problem getting the system to boot after an install (screen generally just goes black), this could be something to look into. I don't know about Sony's, but I had this problem a LOT with Dell laptops.
The situation with nvidia is a lot better now. Ati is slightly more problematic but still easy enough.
Geting restricted drivers running under fedora etc...is still a bitch. Ubuntu is easy peaty.
Oh, that's great! I didn't ever have a problem getting the drivers installed, just the initial getting Ubuntu to boot up properly with the included nouveau driver. I always had to start in a console session and rip those drivers out just to get it to boot up. After that, getting the restricted drivers installed was no problem.
Haven't done it with a newer Ubuntu though. I've been on 10.04 LTS, as I HATE upgrading everything every 6 months, lol. I just don't have the time to spend fussing with it (otherwise I probably would though :D).