If it's under warranty
If it's under warranty still, sounds like a great time to exercise that. Odds are it's an issue with the logic board. Good news is that it isn't likely something that's the harbinger of something bigger. Bad news is that there's no way to fix this short of a logic board replacement. You could buy a USB sound card, but that would take up a valuable USB port on a Mini.
So, you can either decide to live with it, get a USB sound card, or replace the logic board.