In the above responses shows all the major issues keeping all this from taking off, which is a pity:
The CPU grunt;
The ability to run windows OS (and thus associated astronomical instrument software) is paramount;
The cost.
As mentioned fit2PC...well out of the league for most users. The others mentioned...cost is within the target zone but they use other OS thus the lack of support. Hi Matthew...I haven't even considered something like that....but it's worthy of consideration. Expandability with it might be an issue...lack of USB and GPIO pins.
As Rick said....USB over Ethernet will be very nice but it's not the full blown panacea. Something that runs Ethernet, wireless, RS232 and USB...now that's the ticket.
But which one?
Difficult to say...probably Pi is a good place to start but those other mini boards with RS232/wireless/BT inbuilt would make it a hell of a lot easier.