if you only want it to stream 1080P video to a single device with no transcoding, you can do that with a 5 year old $300 laptop lmao. (which if bought now could probably be like $100 or less).
i mean any old $50-80 used dell pcs would be fine with the core 2 dou/core 2 quads.
if you want to stream to a single 1080P device, but that device cant naturally decode the video (most phones and computers can do it easily. many consoles and set top android boxes can. i know roku cant, think amazon fire can), then you would be fine with like a pentium G3258 build. (even that is overkill for just a single transcode stream. a core 2 dou could still probably do that if at 3.0ghz)
if you want to stream to multiple devices which you need to transcode video for, you will need a much better cpu.
my home media server is using an i5-2500K, and it can handle 4 plex transcoded streams to 1080P simultaneously. my home media server i was able to get for under $400 using the 2500K, 16GB of ram, P67 motherboard, 4TB HDD, 120GB SSD, 500W xfx psu, and a tv tuner card for getting antenna tv to be streamable/recordable.