Creating a Backup Device for Time Machine

I have wiped data from a high volume external drive, and also have a Mac mini that I have wiped all data from initially to recycle. So I would like to use the mini for my backup disk but Time Machine doesn’t recognize it as a storage option when I click on Add Backup Disk… What to do? Any questions or work arounds you can provide would be so helpful! I am using a MacBook Air and have connected the mini to it.