OP:
For a 2014 Mini, you don't need the Samsung t7, a t5 will do.
But the t7 will work.
Actually, I suggest you
get one of these for $9:
SABRENT 2.5 Inch SATA to USB 3.0 Tool Free External Hard Drive Enclosure [Optimized for SSD, Support UASP SATA III] Black (EC-UASP)
tinyurl.com
Then, get a 2.5" SATA SSD, 1tb in size.
You DON'T have to spend a lot and buy top-of-the-line.
Here are a few:
Ever wonder why your phone responds faster than your computer? It's because your phone runs on flash memory. Add flash to your laptop or desktop computer with the Crucial BX500 SSD, the easiest way to get all the speed of a new computer without the price. Accelerate everything.
www.amazon.com
Specification Capacity: 256GB/ 512GB/ 1TB form factor: 2.5 inch interface: SATA 6GB/sec performance: Up to read 550MB/s, write 520MB/s * Performance may vary based on SSD capacity, host hardware and software, operating system, and other system variables. Mtbf: 2, 00, 000 hours TBW: 256GB: 200TB/ ...
www.amazon.com
Inherited from GX2, the TEAMGROUP AX2 2.5” SSD has features of low power consumption, high-speed transfer, etc. The SLC Caching technology makes the read/write speed of AX2 4 times faster than traditional hard drives. In addition, the Wear-Leveling and ECC can enhance reliability and prolong serv...
www.amazon.com
NO WAIT BOOT-UP, BETTER PERFORMANCE, RENEW YOUR PC WITH INLAND PROFESSIONAL SDD Fast start-up, loading and file transfers Get up to 500MB/s sequential read speeds for no-wait boot-up, shorter applications load times and quicker data transfer. Incredibly load and run games, videos and photos in an...
www.amazon.com
Wake up your PC's true performance potential! Engineered in the USA, the leverages the latest advances in 3D TLC NAND Flash technology to provide an ultra-fast, robust and reliable storage solution. - MEDS (Mushkin Enhanced Data-protection Suite) ensures valuable data is protected - L...
www.amazon.com
Then...
Just drop the SSD into the enclosure and snap the cover on (no tools needed)
Boot the Mini from the external drive that worked before
Connect the SSD in the enclosure
Use disk utility to erase it.
You didn't tell us which version of the OS is on it
If it's High Sierra or older, use "Mac OS extended with journaling enabled, GUID partition format
If it's Mojave, use APFS, GUID partition format.
Erasing will take only seconds.
Now, quit disk utility.
Next, you need CarbonCopyCloner. Get it here:
www.bombich.com
Open CCC
You will see three "boxes" in the middle
On the left, put your source drive (the external drive you're booted from)
In the middle, put the SSD
Ignore the box on the right -- not necessary.
Then click clone.
If CCC asks if you wish to clone the recovery partition, YES, you want to do this.
Cloning will take a little while.
When done, you can quit CCC.
Now, you want to test the new SSD to see if it will boot.
Restart and IMMEDIATELY hold down the option key and KEEP HOLDING IT DOWN to invoke the startup manager
Select the SSD with the pointer and hit return
MOMENT OF TRUTH:
Do you get a good, fast boot?
If so, open the startup disk preference pane
Click the lock and enter your password, then click the icon for the SSD
Close startup disk pref pane
Now...
POWER DOWN, ALL THE WAY OFF
Disconnect the original boot drive, so that only the SSD is connected
Now, press the power on button
If everything is working, you should boot from the SSD and be taken to your login screen