I have a similar problem, and hopefully some ideas for you:
I recently killed my Asus AT3IONT-I motherboard with a 120 Volt short... this was a great MB for my HTPC. Fanless, enough power for full HD over HDMI, optical digital output etc.
It's hard to find a replacement--at least I cannot find one that's fanless and has HDMI. There are some Motherboard / CPU / VGA Combos out there with CPU fans, which could be replaced for a quieter one. The ones from Gigabyte appear to be well received. However, getting a mini-ITX w/o CPU and adding one is not necessarily that much more expensive, so also check that option.
In terms of OS, it depends what you want to do with it. I used mine only for Music and Videos, and running XBMC on it was just perfect for that. I used the XBMCbuntu distro, it's basically Ubuntu revamped to run only XBMC (right at boot time). If you can get your hands on a remote-control dongle for the original XBox, you can modify that to plug it into a USB port on the HTPC and thus get a remote-control. I did that, it was fairly simple to do (and cheap...)