Essentially, get the cheapest thing you can with an 8th gen or newer intel i5, and put decently fast storage on it. 8th Gen i5 comes with intel quick sync which is dedicated video encoding and decoding hardware.

It’s not exactly mini however I think there are small form factors available, just bought a HP elite desk 800 G4 for £100, and found a good deal on eBay for a decent name brand 2tb SSD (probably overkill to use an SSD but I don’t want noise).

You don’t need an i9 and a 4090 for jellyfin.

This was a good video on it (in my opinion) https://youtu.be/WCDmHljsinY?si=_HAXr1BEusdxeTYW