best external backup for Mac Mini?

For desktop computers like a Mac Mini, I only use quality 3.5-inch hard drives in an enclosure that has its own power supply. I buy these enclosures:

https://eshop.macsales.com/item/OWC/ME3NH7T00/

and install SATA 6G 7200 rpm Western Digital Black series drives. A solid-state drive (SSD) is overkill for Time Machine backups. Once the initial backup is made, I never sense the OWC externald working, but they do. Very unobtrusive and reliable; we have one OWC external that is over 15 years old and still works.

Many of the little bus-powered drives (must get all power from the computer) like those that go on sale every other weekend were designed to do just that: go on sale every other weekend. They are too often less reliable in practice.

Yes, the OWC/WB Black solution cost more but it always looks attractive when I ask myself, "How much is my data worth?"

IMO an external 2.5in drive make the most sense. You should get one that is a bit larger than needed to hold the data on your Mini.

To backup, the Carbon Copy Cloner program is perhaps the most popular method. You would need version 6.

CCC will allow you to boot from your backup so the external should be an SSD not HDD.

LaCie, G-Drive are popular brands. Examples:

https://www.amazon.com/LaCie-Portable-External-Solid-State/dp/B092RCKNPJ/

another

https://www.amazon.com/HP-Portable-External-5MS30AA-Black/dp/B08GV9S2JC/

However one should note that the brand of the unit is rarely the maker of the actual drive.