Hi William,
Are you in the US, using a 110V system?
I have not disassembled a Mini but it is not uncommon for switching power supplies to capacitively couple one of the AC power legs to case ground. This may explain the tingly sensation, especially if you are also touching another piece of equipment that is grounded. I am not aware of any "factory approved" grounding method.
You must rule out all other equipment as its cause. Disconnect everything connected to the Mini as I suggested earlier. If the problem disappears look elsewhere for its cause.
I do not know why Minis use a grounded connection while iMacs do not, but capacitively coupling the case reduces the Mini's ability to contribute to electromagnetic interference. I presume this is to comply with the many certification standards to which the equipment is subjected.
Some time ago I responded to this thread in which the OP wanted to find a wired, non-optical mouse for EMF sensitive user. Klaus1 suggested a link to a site (lessemf.com) that is not working now. Perhaps that site, or something similar, could be perused to find an elegant way of grounding the case. Drilling and tapping a hole for the purpose of bonding its case is not something I would do to a Mini.
You must first be absolutely certain you do not have another problem that is causing this. As I suggested earlier, you must be 100% certain that all your house wiring is correct, and that the Mini as well as all the equipment connected to it is working properly. An improperly wired system can result in circulating currents in what you think is "ground". This can cause problems that will drive you up a wall.
If something is amiss, grounding the Mini's case will only hide a symptom and could cause it or other equipment to fail in dramatic fashion. I had a ground fault once, the manifestation of improper wiring in a rapidly expanding office. It resulted in $15,000 worth of damage. Leaving the Mini's case "floating" with respect to ground is safer from that perspective.