Salix just came out with it's live edition of XFCE. It's a Slackware-based distro, but one that has a nice XFCE desktop and some extra tools to make setting up easier.
I have an 11.6" laptop with a Pentium CPU and I run Slackware with Openbox and Tint2 (so basically like ArchBang, which you may also want to look at). It ends up being something like having a Chromebook, but with all the XFCE apps.
I have the same computer and Lion isn't as good as SL was on it. If you have the install DVDs I would first try to boot from the DVD by turning off the computer and holding cmd+c on boot (quickly throwing the DVD in there onstartup if it won't stay in the machine when on.
But before this it's worth trying to use disk utility to restore disk permissions, verify the disk and repair the disk. You might need to run these programs several times and even when it you get the green text stating everything is OK, do it one more time - errors can show up on the next pass.
Another tip is to download and burn a Linux Live CD (Salix for example - on CD as the CDs seem to stay in the drive) and use the linux tools (it's also called disk utility in Salix and most Debian based distros, like Mint) to check the health of the HD and do whatever else - I've found this to work better than OS X's disk utility in giving feed back.
Finally, some sage advice: if you run repair disk, repair disk permissions etc and get an error message, write it down or copy it and google the shit out of it. It will tell you what the problem is and I guarantee you will find a forum that has an answer to your problem.
I read about a similar problem from the Arch reddit, which lead me here.
I run LMDE without Flash, and have not found a downside yet! (I guess not watching Starcraft tournaments is one downside :)
Didn't mean to sound stupid, it's just that different people use the same terms to mean different things.
These days the go-to answer to your question seems to be Arch and I can't argue against that, but if you want to go off the beaten path I see that Crux just released a new version.
Salix is Slackware-based but has some decent configuration tools, uses gslapt for packages and has a version of Sourcery for building additional packages.
Those computers will run anything Linux, the only problems you may have is in finding drivers but I doubt that these days (it's a problem that has mostly disappeared, as Linux devs have worked hard to make it a thing of the past). A favourite youtube channel of mine is SneekyLinux, he gives a run down of many distros as they are released, look back through his video history to see Mint etc as recommended above.
I hope you enjoy Linux, the worst thing about it is convincing people how good it is. It can be technical, but also easy to use (easier than Windows imo). A favourite of mine for no particular reason is Salix Linux with the xfce desktop. Something about it just worked for me.
This how to works for me. I did it so I can get everything to go through my DAC.The site is Salix, but I've also used it on Arch and a couple of other distros.
Yeah or Salix If you want a window manager on top of Slackware from the getgo . But hey I am more of a stupid guy, when it comes to "advanced" linux. So that is why I personally would start with a Linux that has a window manager as a start.