MacStadium does remote Mac mini hosting, offering customers a fully connected Mac mini in a secure, controlled datacenter (just like the folks at MacMiniColo, who we've visited and written about before). If you have a Mac mini already, you can even send it to them for hosting, or you can rent or buy one yourself right from the company. Having a remote server all for yourself can be very helpful, and using a Mac mini for the job can be one of the cheapest and easiest ways to do so.
In fact, not only does MacStadium run hosting, but the company has also been putting up blog posts over the last month or so, going through a few different ways to make use of a connected mini for yourself. You could install a remote Minecraft server on it, for example, creating a world that you and your friends can log into anytime. Or you could set up something like Kerio Connect, which serves as a Microsoft Exchange-style way to share and send email, contacts and calendars across a number of different users.
Obviously, these tutorials are for MacStadium's hosted minis, but they should work for any Mac mini you've set up to be accessed remotely. Apple's littlest Mac is relatively cheap but still very powerful and MacStadium's tutorials could be a start to help you get the most out of your remote mini.