It is possible to put a 12.5mm drive in a Mac Mini, but it is a project and requires a Dremel tool with small grinding bits. We just finished doing one.
There is enough room, but the Mini case has four plastic risers along the outside edge that lift up a 9.5mm drive to the right height. You need to shave these risers down enough to fit the drive in, but not so much that you sever them entirely, as they double as case supports. The most important risers to shave down are the ones nearest the connector. These need to be brought down flush with the metal frame. That will probably sever at least the front most one. You can leave enough plastic on the riser furthest from the connector, but that means you will need to enlarge the hole for the screw on the side.
It is a tight fit, and my only concern is that this may reduce necessary space for cooling air and could lead to overheating. Well, time will tell!