What do you consider a minimum budget?
Also, we need more details about what you want/need:
Bitstream HD audio?
Record live TV? Antenna, cable, satellite?
Store a large library of media files? How much storage is needed?
Do you want it to look like A/V equipment?
Do you want to control it with a remote? Wireless keyboard?
Extreme silence needed? (for example, in a bedroom)
Do you have a Windows 7 license or are you willing to try Linux?
Any parts you already have to use in the build?