Raspberry Pi is the most energy efficient and easy. Works perfectly if your clients can direct play the media. If transcoding is needed then you need a Celeron based NUC with QSV support (Intel Quick Sync). One of those can handle transcoding for a few clients unless we're talking 4k source files then it's like 1-2 clients max and no tone mapping. Next step up and we're talking something that can't really be called energy efficient.
Make sure the clients can play the media without transcoding, there are lists of formats Jellyfin support natively but generally speaking H.264 in a .mp4 container is a safe bet and pretty common to find. If your files aren't in that format then you can quite easily remux them, and add some automation to it so it's done ahead of time. The Pi can remux files but it will be pretty slow so best is if you can have another computer do that part.
Oh yeah, if Jellyfin insist on transcoding it might be due to poor / shaky connection such that it wants to transcode to say 480p / 360p or something and then best is to try and turn it off completely in settings.
I believe Jeff Geerling talks a bit about this on his video on Jellyfin. If memory serves, he has some way of preparing his files that I guess you could describe as pre-transcoding.
Honestly any mini PC made in the last decade should work. I would go on eBay and look at used machines. If you can go for something made in the last 5 years that has expandable storage.
Depends on your budget and use case. Jellyfin’s “Hardware Selection” documentation has you covered as to general specs, so maybe consider them a shopping list.
Briefly, a mini PC with the following will work for a “normal server:”
CPU: Intel Pentium G4560, Intel Core i3-7100 or newer Pentium or better
RAM: 8GB or more
Storage: 60GB SSD storage for Jellyfin files and transcoding cache.
Graphics: Intel HD 6xx (7th gen integrated graphics) or newer, Nvidia GTX 16 / RTX 20 series or newer (excluding GTX 1650). Intel is recommended over Nvidia. AMD and Apple Silicon are not recommended.
If you don’t need transcoding, or can use low power transcoding, Intel 12th gen or newer Atom CPUs with integrated graphics will serve.
I’ve run Jellyfin on a R Pi 4 but experienced glitches on playback with a Roku with media that required transcoding. It wasn’t awful, but knowing myself, I knew it’d get old quickly so I went back to MiniDLNA which works a treat when playing back on a mini PC with VLC or Kodi.
I use an M1 Mac mini. It might be a little overkill, but I use the mini as a server for many things other than jellyfin too. encodes are lightning quick with no fan noise or heat and very low power consumption.
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).
I use a usff Lenovo thinkcentre m97p with external hdds for media. It has 4th-gen i5 (2 cores) and 8GB of ram. If I don't use transcoding, it's perfectly capable. For transcoding it would depend on the media :) I only stream fullhd.