Realistically unless you're up for spending at minimum of a couple hundred bucks, you're not going to get a true mini-pc (unless you want to buy a NUC or something, but that kinda defeats the purpose). Any mini-itx case you'd want (<14L) is going to be around the $100 mark on the cheap end. Unless you really luck out and find a compatible used motherboard on the cheap, youre looking at another $100 minimum. Could get a used 4600g for ~$65, but it's probably not going to come with a cpu cooler if it's used. Add in a power supply that fits, that's another $140 on a good day.
Small form factor PCs are stupid expensive because it's a fairly niche market that very few people absolutely need.
What I'd personally do is find a used dell inspiron or similar for <$150 and toss that all in it