USB Optical Mouse not working with Mac Mini
Best way to see if they recognize each other:  Go to System Information.  You can do this a couple ways, but the quickest is to hold Option and click on the Apple icon at the top right. You will see "System Information" where "About This Mac" used to be. 

On the System Information screen, along the left you will see these categories: Hardware, Network, and Software.  At the bottom of the Hardware portion, you will see the USB section.  Click here, and you will see your mouse if it is recognized by the computer.  It will be listed under one of the USB Hi-Speed Bus's.  

You've already taken the other troubleshooting steps, so I'm guessing that it won't be there.  If that's the case, then the mouse doesn't work with your Mac, unfortunately.  I have only seen this on one other mouse, and it was a cheap Logitech.

I hope this at least confirms something for you, even though it may not be the news you wanted.  Might have to get a new cheap mouse, but at least they aren't too terribly expensive these days.