A lot of people seem to be fond of little devices like this Lepai Amp on Amazon
. I have something similar that I picked up for $8. There are thousands of cheap little amps like that if you look around, most simply need a 12VDC wall wart or other power source. Not the highest quality, but after looking around at tutorials on building my own with a BOM in excess of $50 for a simple pair of PC speakers, I knew there had to be a different way. The one I particularly like right now is this Drok 5v amplifier
. It isn't going to rock the house, but it is very tiny and you can power it with a USB cable. A bit more DIY but I figured I would throw that out there.
Something like an Intel NUC should be more than adequate for your needs. Low power, low or no noise, very compact. Then add a USB Blu-Ray drive and you should be set. You could also build your own small form factor PC. Anything else pre-built will likely be too cheap in quality or too expensive in my opinion. If you want cheaper, then there are a bunch of Android TV boxes and the Apple TV. The old Apple TV is on sale since the new one should be out in the coming weeks. Same with the Google Nexus TV player, there should be a new one due this coming season and the old one can be found very cheap. So if you have people familiar with either Apple iOS or Google Android products, those would be about as simple as you can make it. Of course this would need a stand alone Blu-Ray player which makes things complicated.
Do you have a cable for the projector running to the point where you want to use the machine? If you don't and need it to be wireless, it seems like stuffing a small NUC/TV box by the projector and using Bluetooth for your keyboard and mouse would be less problematic than trying to wirelessly stream video. I know it can be done and for many people here it is a no-brainer, but if you want something dead simple then this is certainly an option.