I would upgrade to at least a late 2014 Mac mini; older minis will not support Catalina or Big Sur, so if you're going to spend money to upgrade your mini, make sure you get something that supports current macOSs AND which gives you an upgrade path. If you get an older-than-2014 Mac mini, you'll only be able to upgrade to, perhaps, Mojave, and then you're stuck again with no upgrade path and no support. I'm running a late 2014 mini under Catalina, and I can still upgrade to Big Sur if Apple ever gets all the bugs worked out of it.
That said, Catalina and Big Sur changed the file system from what had been being used to APFS, which if you check some of the threads in this forum, you'll see that APFS is better suited to (preferably) SSD drives or to (acceptably) high-speed spinning platters (e.g., 7200rpm). Performance will suffer somewhat if you upgrade to Catalina and keep it on a slower hard drive, such as the 5400rpm drive that is used in the late 2014 Mac mini. I run Catalina on a G-Tech external drive which spins at 7200rpm, has 64GB of cache, and is attached to my mini with a USB3 cable. Performance is decent.
I went straight from El Capitan to Catalina, and I found the process non-trivial (as Cory will attest). If you upgraded to Mojave, you would still be under a supported release for probably another year, and you wouldn't have to worry about APFS performance issues. And, once support is dropped, your system will still run, it just won't be updated with Apple fixes. I ran for a couple of years unsupported on El Cap, because I didn't need any of the newer features - until recently.
Cory's point about your software inventory should be heeded; if you have to pay for all new software or licenses, you may as well go whole hog and buy a new mini.