Since a home media server is left on 24/7, I strongly recommend buying either a NAS or a laptop for this purpose. Laptops typically use less than 5 Watts at idle with the screen off vs about 30 Watts for a desktop. 30 Watts under load (non-gaming) vs 100 Watts for a desktop. It'll be even higher if the desktop has an add-on GPU card, instead of integrated graphics. NASes are made using laptop components and have similar power requirements (their biggest power draw turns out to be the HDDs, since they use the larger 3.5" HDDs). About 5-8 Watts idle, 15-20 Watts under load.
By coincidence, if you pay the U.S. average of 11.5 cents/kWh, then each Watt used by a device left on 24/7 for a year translates almost exactly into $1. So a 30 Watt device will cost you $25/yr more than a 5 Watt device. A 100 Watt load will cost you $70/yr more than a 30 Watt load (why I went with a laptop for my security camera system). So in a few years, a desktop used as a home media server can end up costing you $100-$300 more due to electricity cost. More if you live in a state where electricity costs more, like California.
For the same reason, you're usually better off with a newer system than trying to re-purpose an old system. Although the biggest reduction in power consumption came with Sandy Bridge (2nd gen Core i processors). Any processor Sandy Bridge and newer should be suitable, though the newer gen processors should save you $10-$20 per year. If you plan to use this for long enough, that savings may warrant just buying NAS with more modern components.
If your router has a USB port, a cheaper and lower power alternative may be a USB HDD plugged into the router, and something like a Roku or Chromecast plugged into your projector (though these are usually HDMI out). Your router is just a low-power PC usually running Linux. The Roku or Chromecast can stream from network shared storage, and handle the video decode.