When choosing a Linux distro to install on your PS3, your options are slowly cut down compared to if you were just installing on your own PC. Linux platforms have to be specifically coded to allow the OS to handle all 8 of the PS3's CPU cores. And no, I'm not exaggerating. The PS3 really does have 8 powerful cores to work with. Given that your options are slimmed down, what do we have to work with?
Well first up is Ubuntu. This version of Linux is pretty much a household name nowdays. And even people outside of the IRC channels recognize the name as a PC OS. Right up to version 9.10 have been ported and made available for PS3 users. If you're a complete newbie to anything Linux, then Ubuntu is probably the best place to start. It is easily the most user friendly Linux distro out. It even comes pre-packed with things like Pidgin (MSN, AIM), Mozilla Firefox, and Open Office if you want to work on your PS3. Given that these pieces are included free and are ready to go right out of the box, it's a Linux system that can be run straight out of the box.
Your next option is Yellow Dog Linux. Yellow Dog Linux was the very first Linux distributions to be made available for the PS3. Sony actually commissioned Yellow Dog to develop on top of the PS3 framework from the ground up. So there is no porting issues or problems with any of the PS3 hardware. However be aware that if you own a standard TV then you will have a little bit more trouble then usual. Yellow Dog was developed to be natively run on HDTV's only. That is not to say you can not run it with your old box, but there is just a little more work involved. Much like Ubuntu, this version of Linux for the PS3 comes with many home comforts like Firefox, Open Office and Gimp.
Your last two options (Although not really options), are OpenSUSE and Fedora. Really you should not bother with these two unless you really want to experiment with using your PS3 as a web server. Fedora especially had issues with recognizing a lot of the PS3's hardware and Ports. And up until recently it was rarely able to use USB ports. These two are really only applicable if your an absolute pro at Linux.
Like everything in technology, the Linux family for PS3 just keeps getting bigger. And more then like by the time you read this, there will be a tonne more Linux distro's for your PS3 to choose from. However, no matter what comes out, I would always stick to either Ubuntu for its ease of use, Or Yellow Dog for its native PS3 usage.