Hard drive not showing up on Mac Mini A1347
I had exactly the same problem (on an iMac). And no, if the system doesn't see it, you can't do anything to test it. Opening the computer is the way to go. I opened mine and discovered that one of the Sata connectors (my Sata cable had two connectors : data and power) had come lose. It might be something similar, although most probably I'd say disk is toast. 

What you can do once you open the computer is put your disk in an external enclosure and test it with another computer to see if it's recognized and properly formatted (even if it's not properly formatted, it should show up, but as long as it's in the enclosure, why not format it correctly if you see it's not). That will save you the trouble of closing the Mac mini after unplugging and plugging the Sata cable on a bad disk. If the disk is fine, check the Sata cable and Sata connectors, and try again. If you computer still doesn't recognize the disk, you probably have a bad Sata cable. Could be a bad Sata connector on the motherboard, but that's unlikely.