For hosting a Minecraft server (or any server, really), you need pretty high upload speed.
In Canada, I have a 30mbps (down) connection with only 1mbps upload...
Check with your friend first if his internet connection can even support hosting.
That said, for system specs, the recommended system requirements are quite low, so (assuming that your friend doesn't want an upgradeable build), I would go with
i3-2120 (~110$ on newegg with rebate) maybe even consider Pentium G?
4GB 2x2GB Mushkin Enhanced/G.Skill/Kingston memory (could be budget RAM) (~25$)
H61/H67 full/micro ATX (depending on case) (~85$)
500w Corsair/Seasonic/Antec/XFX/Rosewill PSU (Rosewill Green 530w on special, newegg ~50$)
AMD Radeon HD 7850 (~250$) or downgrade to fit in budget
Seagate Barracuda or WD Caviar Blue 500GB (~85$)
ThermalTake V3 Black Edition (~35$ after rebate on newegg) OR microATX case
Optical drive (~20$)
Total: ~660$ without shipping/tax
IMHO, that 60$ more is justified, but if it's a fixed budget, it's a fixed budget...
Never played minecraft but modern games should run fine on this build, so I'm assuming minecraft isn't a more demanding game. (it isn't, right?)
EDIT: forgot optical drive