Are you going to be running planetarium software?
If not, then any basic PC will work.
For instance:
- Intel i3 CPU, whatever is cheapest
- Cheap mobo, any brand
- 4gb of RAM (cheapest)
- case that looks good for you
- 400W~ PSU
- 1TB HDD or whatever size you want.
Should be able to build that for $300-$400 or so with cheap parts.
An atom processor machine will work fine as well if power consumption is a concern.
If you are planning on running planetarium software that is image heavy, such as Starry Night pro, then you will want to add a graphics card to that.
An AMD 7750 for around $100 will be good enough.
PC's dont need to be very powerful for astrophotography. Only planetarium software is really power hungry, except Cartes Du Ciel which has no graphics
If your after a prebuild PC then take a look at a cheap Dell or system built by a PC store. Dell do make some really cheap basic machines that will work well.
If your after a tiny PC to throw in a corner, you can get some prebuilt and barebones from this company:
http://mini-box.com.au/barebone.html
http://mini-box.com.au/complete%20kits.html