In your budget, your plex performance would be mediocre at best. I know it. Because, Plex is not designed to run optimally for the CPU architectures mentioned here. It will run, meaning the software would function very well. Just your movie playback experience wont be good.
Not my words. So not looking for an argument with anyone. And I happen to run Plex with a i5 NUC now, for my 4k playback, it just sucks. Stutters, frame drops etc.
As many would have said, R Pi is a horrible suggestion. Stay away from it. So are synology low end versions. The high end versions are way too expensive to be cost effective.
Whichever option you go for, look for the
passmark score of that CPU. The higher the score, the higher your plex transcoding performance would be. This is my plex server soon, costed me around ~520USD. With taxes. Shipped to my home. Just need to assemble it. Been sitting around for 4 weeks in my shelf.
2 * E5-2650v1 8-Core CPU - Used
2 * ARCTIC Freezer i11 CPU Cooler - New
Gelid Solutions GC-Extreme Thermal Compound - New
Supermicro X9DRi-LN4F+ Motherboard LGA2011 - Used
24 * 4GB ECC DDR3 RAM - Used
512 Crucial SSD on StarTech M.2 NGFF SSD to 2.5in SATA Adapter - Plex metada, Already had
Cable Matters 24-Pin ATX Power Supply M/F Extension Cable - 12 Inches- New
StarTech.com 4x SATA Power Splitter Adapter Cable - New
6 Pack Straight SATA III Cable 6.0 Gbps 18 Inches - New
PSU - CORSAIR 550W 80 PLUS GOLD Semi-Modular- New
Case - Phanteks Enthoo Pro- New
6' EPS 12V 8 pin to Dual 8 pin Y Splitter PSU Motherboard Power Cable YEP-S828.- New
The idea was to get the best passmark score machine.
This was an expensive ordeal atleast. Only went for this because I was about to go for a i7 Synology 8 bay, had the budget and have lots of 4k High bit rate movies.
There are lots of single Xeon cheap options. I happen to have one more build coming up as well, already have the parts. Costed me like 180USD in total, which happens to be in your range as well.
EDIT:
http://www.ebay.com/itm/SuperMicro-X8SIL-Motherboard-w-Intel-Xeon-X3470-8GB-RAM-I-O-Shield-X8SIL-CT033/112394975698?ssPageName=STRK:MEBIDX:IT&_trksid=p2060353.m2749.l2649
This is the board for anyone that is wondering about the cheap build. It is a Xeon processor, Motherboard AND 8GB ECC RAM combo. You can offer your price. But it is usually with much less of a diff from what you see there.
Couple that with a LSI 8i card like this,
http://www.ebay.com/itm/112327269957 and whoa... you are in for a treat. You can offer your price here as well.
Note: Lowballing both of those items are of no use. Try your luck though. These combos been popular in the plex communities. So they know these things move.
On a budget.