In trying to reinstall OSX to our Mac Mini, the computer accepts, reads and then rejects the disc, nothing but blue on the screen. What should I do next?

Question

Our 2009 Mac Mini became quite sluggish and then just wouldn't function.  I tried reloading the OS with the original disc from Apple, but the computer rejected the disc and the screen was all blue.  Next, I shut down the machine, inserted the disc, turned it on, the screen is a faint gray in color, Apple logo in the middle, the disc came out.  What should I next try? 

Answer 1

Apple logo in the middle,
The Apple logo is not contained in your Mac's ROM. Seeing it indicates that your Mac found a bootable device, and loaded the initial blocks from that device. I expect you booted from the damaged hard drive.
Use the Option key held down at Startup to get clarity over exactly what drives can be seen, and what your Mac thinks they contain. Wait for it to draw an icon for every potentially-bootable Volume. Deliberately select the device you want to use as the Boot Drive.
You must have a wired keyboard plugged directly into the chassis to use snag keys at Startup. A wireless keyboard that is already paired can be used, but the exact timing is more difficult.