I think a RasPi B model would make a perfect cheap, low-power home file server. I want to use one and a couple HDDs to set up a low-cost network storage solution. HDDs would obviously be USB or use external USB enclosures, I guess it depends on which ends up being cheaper.
I plan on installing SAMBA to share files with my Windows PCs and probably a FTP server as well. I'd also need some sort of efficient torrent client (rtorrent maybe?) that would be capable of downloading at high speeds on the RasPi. I wonder though if it would be capable of downloading torrents at about 10MB/s, since both the HDD and NIC would be using the same USB port on the CPU. That should still be less than half of the theoretical USB 2.0 bandwidth, so I hope it'll manage that without too much trouble. Any ideas if this should pose a problem or not?
Along all these things I'd also have to have some way to remotely access it, either by SSH (but then again I'm not very familiar to Linux) or by some flavor of VNC.
Something that could wake up other machines on my network remotely would also be pretty useful, but I don't know if anything like this has been made or not. If not, I think I'd be up to the task of writing it myself.
Other than that I was also thinking of running something like PS3 Media Server on it, but even if Java runs properly (afaik there's no official JVM, right?) it probably wouldn't have the hardware resources to handle 1080p live transcoding, worth a try though.
Anyway, I'm really excited to see the RasPi release and I'm sure I'll have lots of fun toying with it.