late 2012 HDD 2.5 I change it to SSD. Is HGST the type as I want a case for the HDD as a Time Machine drive

HGST - is brand name (Hitachi Global ......) and it really does not matter what HDD (2.5") you use.

More likely, that HGST would be the one you removed from Mac Mini.

Given the age of that 2012 MacMini's HDD -- I will be cautious to use that drive as Time Machine drive.

Most of the 2.5" --> USB 3.0 case would work. However, due to many choices and selections available for the HDD enclosure, I would to get it from OWC - they have good warranty, juts in case any issue might arise.

Additional note - all the drive will fail -- does not matter 2.5" or 3.5" HDD for Time Machine. It will be wise to back up frequently and have multiple backups (multiple TimeMachines).

If you want to "upgrade" your Mac to an SSD, use an external SSD to startup from: How to Setup and Use an External SSD as y… - Apple Community.

If you want a Time Machine backup drive, a G-Technology G-DRIVE mobile USB-C Portable Hard Drive - Apple would work well.

Jack

1. The deed is done. The OP has already swapped out the stock HDD with an SSD and is looking to re-use the HDD.
2. Plus that over priced G-Drive from Apple is USB-C and a Late 2012 Mac Mini only has USB-A 3.0 ports.

+1 adding to what samtenor has shared.

The old Mac Mini's internal drive has a lot of miles on it and may not be all that reliable.

Personally I would put it into a 2.5" external USB 3.0 enclosure and keep it around for redundant Clone backups using CarbonCopyCloner. https://bombich.com

Then I would encourage you to get a more reliable external drive to use for Time Machine.

for some examples, see > https://eshop.macsales.com/shop/external-storage#drive-configuration:single-bay

You’re welcome.

I suppose I misread the thread. Sorry.

Regardless, the G-DRIVE ships with a USB-C to USB-A cable. 

Good to know. Sorry, I only looked at the Compatibility list and it didn’t list the 2012 Mac Mini.