Lubuntu 13.10 has a minimal desktop environment that is suitable for low-end hardware like your mini PC, however even Lubuntu requires at least 4.7GB of available disk space for best results.

Instructions for installing Lubuntu 13.10 and later on the mini PC

Boot the Lubuntu live USB.

Open a terminal. Type sudo -i leafpad /usr/lib/ubiquity/ubiquity/misc.py. Click on Options and select line number. At about line #853 will be something like "min_disk size = size x 2 #fudge factor", change 2 to 1.4 and then save.

Run the Install Lubuntu. It should now say minimum disk size 3GB. Just be sure when you get to the disk partition/install part select Other and set up the drive as one ext4 partition as /. No swap; a page will come up saying you didn't make a swap partition, but just ignore it.

These instructions also work on Ubuntu and other *buntu distributions that also use the ubiquity live CD installer. In Ubuntu the default text editor is gedit instead of leafpad.