Hmm, well... hidden is not an issue, really, though it has to be an HTPC style... noise isn't my biggest concern either, loud machines don't seem to bother anyone in the family. It does have to be relatively cheap, and it does have to have an HDMI port.
Personally I would build either a miniITX HTPC or a microATX HTPC. The microATX one would be a bit larger, although you might be able to build it cheaper, if only because the case might be cheaper. To be honest I don't think there is a lot in it cost wise.
If buying miniITX, make sure you get a board with nVidia onboard (should have HDMI, or at least DVI). With a microATX HTPC, you'll be able to fit an nVidia graphics card (providing the case is large enough), or just get a mobo that has nVidia onboard.
What bits do you have laying around? That's usually a good place to start for a cheaper machine!