Reading the title, I was going to suggest seeing how much space your log files are using. But, you've done that, and it didn't do anything, because your var is on it's own partition.
Your root is on a ~1gig partition. It's full. Multiple partitions are sometimes good and sometimes bad... If your log files went crazy and filed up var, you should still be able to run the system to fix the problem. In your case, you have a partition scheme that's a bit out of wack. It will take a bit of work to correct it... You can edit your partitions to move some space to your root partition. Before do that, be sure to back up your data. I would say chop var down to 1-2 gigs and move that freespace to root.
You could also add a little more to boot, or delete out older kernels that you don't ever run. Keep the one you run daily, and one of the previous best versions.