I suspect this is not so easy, and needs quite a bit of time and expertise. A fun project for sure but refining it enough for family use might be tricky.
You can start with one of the small-form-factor multimedia PCs on the market, with remote and LCD read-out, but you need to make sure you can find Linux drivers for these features.
VIA EPIA Mini-ITX is as small and quiet as you can get, but there are issues with Linux drivers for video and MPEG (DVD) decoding. See
If you start with a standard Micro-ATX and add components, with quiet fan and power supply you need to figure out how to add a remote. One place to start looking is www.lirc.org.
Or maybe there are commercial products you can use these days.
Best thing would be to find an already successful example, and copy or improve it.