I had thought about this myself....debating between getting another 360 to have extenders at all my tv's and using the desktop to serve them, or throwing together a cheap low power silent HTPC to serve the extenders and leave the main desktop out of it.
Xbox...you dont need a hard drive equipped one, so pretty much ANY of them will work. At near $100 they ran new during black friday, certainly an option at that time. Broken optical drive ones for cheap on ebay and the like is also another option....but I would add make sure it is a slim model! While not dead silent itself, they at least have a bearable amount of fan noise. All the previous ones...much too loud IMO. They are fine for playing games when things are booming and ringing etc as it gets drowned out, but would be pretty annoying for general watching tv and stuff.
Low end HTPC...you can do these for about $200 + hard drive pretty easily.
For example you can pick up this: http://www.newegg.com/Product/ComboD...mbo794210-LM2A
(and click on the link for the computer itself to find 10% off coupon, which works with that combo) + 4gb ram for about $180....then repurpose an existing hard drive since they are so expensive these days or pick up a new one of your choice......now you have a small silent low power full computer for around $200 without much fuss.
Another option I priced out recently...you can get cheap ITX case for $40, decent H61 ITX motherboard with HDMI and other usefull features for <$80, Intel G620 for about $60, 4GB ram for $20 (or 8 for around $30 if you want to run x64)....same deal with the hard drive as above...and there you have a fairly powerfull HTPC for a little bit more then above, albeit with a good bit more work, and having to pay attention to component choices for noise and power draw. Add in a decent passive video card and you will have a machine that can do any HTPC task thrown at it, if you want to move beyond the iintel graphics which are pretty capable themselves.
I think between the 2 cheap HTPCS, besides preference of course, it comes down to if you are an HD netflix user or not. The G620 and the like is going to be MUCH more powerfull then that AMD system, or similar atom/ion systems...but that amd system will play live tv, handle extenders, watch all your downloaded movies, etc just fine, but the one thing it cant do well is HD netflix. The attraction between the 2 is the form factor/noise level/simplicity for the amd system, and of course the power of the intel system.
edit---and one more thing to add as far as netflix goes...you wont be using it on the 360 either unless you have an xbox live subscription which adds $40-$60/yr to your cost there...and it is it's own app on the 360 and not part of the media center on an extender like it is in media center on an actual computer....something else to take into account there if you are a netflix user.