I’m talking about SBC’s, mainly ARM based. (Well, especially about the Odroid, but let me mention other possibilities before it)
So, the RaspPi can be one such solution, but there’s BananaPi, maybe something from Pine64, or Asus(?) tinkerboard comes to my mind.
Intel NUC with low powered J#### CPU could be a candidate too.
But my favourite is definitely the Odroid, namely the model HC4.
I suggest to buy a power meter, which you plug into the mains socket, and it measures how much power your device draws from the powergrid.
I have a very-very similar one to this:
Actually the same, just with european plug/socket.
Having such a meter you can measure the actual power requirement of a device.
If it’s a computer (server?) wait until it fully boots and settles.
Then you will see the power usage during idling.
A home server probably 95% of it’s time will be just idling, and waiting for request to serve. So you can estimate the yearly cost of your device having up and running.
Read the W, and multiply:
E(yearly)=(P(idle) * 95 + P(loaded) * 5 ) * 24 * 3.65 [Wh]
Edit: (Divide the result by 1000 to have a kWh unit)
This is just an estimation, assuming the server will idle for 95% and loaded for 5%. I see on my server, that in 4 days and 5 hours approx 5 hours CPU time is used up, so I statistically consider the idling time was 95%.
Having the value of E(yearly) and knowing the price of electricity where you live, you can make a very good guess what device is worth to buy (if any) and if you buy a more frugal device, in what time it pays back.
So please @Tech_JA, @Sheila_Flanagan, measure, calculate, estimate, and decide what to buy (if anything at all 
).
I’m so much satisfied with my Odroid HC4, that I cannot make an unbiased recommendation, I simply cannot suggest any other device for a home server Please forgive me my highly biased opinion. 
I’m thinking of making a video demo, how my home server performs in my ecosystem, if you are interested.
Of course there’s a negligible chance that the server building project fails, and results in nothing usable. That’s why I suggest to build/config/test-drive a server with a hardware that you already have. So you won’t waste money on a device, which you later can’t use. If you can build up your server on a something-hardware, you’ll be able to repeat it on an Odroid* ( 
) as well.
*: …or Intel NUC, or RaspPi, or BananPi, or…