The Mac mini’s input is a line-level input, whereas the mic line on most analog headsets needs a mic-level input (to power the mic). The mini isn’t giving the mic any power, so it’s not going to work.
You need something to pull the level of that mic line up to what’s known as “line level.” A lot of people use the iMic, others use audio mixers that will provide power to the mic (and other inputs) while providing a line-level output for the mini to accept.
To add to the above advice, it's worthwhile also considering a bluetooth headset directly, if you have bluetooth on your mini
