MiniPCs

I use the little HP Minis which are similar in price to the Lenovo Tinys, and just as easy to find on eBay. It's a pleasure to open them up to do any work in them - great engineering.

I've been happy with Lenovo Thinkcentre Tinys. $40-100 on ebay depending on the specs you need, and surprisingly easy to upgrade if you want. I run a few of them for various home server things (backed up by a NAS box).

Without knowing your system requirements, power requirements, size requirements, future expansion needs, budget etc, it's hard to recommend anything specific.

If your requirements are low and you don't need any expansion slots, how about getting one of those HDMI stick PCs? For instance, you could snag an Intel Compute Stick with a 4c 1.9GHz Atom / 2GB RAM for just $40.There are other variants too with higher specs it you look around. The biggest advantage of these is that they can be powered by just a standard USB charger (typically microUSB), and of course, their size is a major advantage for projects where you may need portability.

If need more power though, then you'd want to get a proper mini PC, like the Intel NUC. Just search for used NUCs and see which ones fit your budget and requirements. Only problem is, if you're buying from eBay then you may need to get the seller to clarify what exactly is included in the NUC, sometimes they might sell it without any RAM/disk etc, or they may not give out the actual specifications of what they're selling. Be extra careful when you're buying a used NUC.

If you can go a bit bigger than a NUC, look for HP EliteDesk 800 SFF or Lenovo ThinkCenter Tiny. These typically have more power, greater expansion capabilities etc, but obviously bigger in size and slightly greater power consumption compared to the NUC.