If you want EASY to install choose Mint (or the like) but if you want stability choose Debian, not that others are unstable, it is relative. Debian will ask you a fair amount of questions while you install, Mint will not. Once you have an install disk of Mint, or Debian, you can go back and try again if you choose the wrong options. If you download the iso be sure to check it with MD5SUM or the like. With WindoZe disks it might cost you a lot of money if you make mistakes.
Be sure the bare bones system's bios lets you use the traditional (legacy) boot option before you buy.