my question is could just purchase the regular mac mini and install server OS X on it?
Responses like this are what keep people from answering your question. Good job.
Hmmmm... Your question, IMO, is ambiguous and unclear. Perhaps you should re-read what you wrote and edit it for clarity.
Nope, in ***hole.
Simple answer - yes...
However, server is just additions to the software giving you a GUI to more easily set up the software that is already there in the client version. Someone who knows their way around the terminal can basically do everything that the OS X server software does.
And, the other (crap) answers are, IMHO, pointing out that adding a hard drive (extra storage) would likely be more efficient than adding a server to your network. But, then that's where the confusing part comes in. Connecting computers through firewire is somewhat limited compared to ethernet, or even wifi - as that's essentially a local connection, and not on your full network.
Hence, though your first post appears clear enough - it begs the question why you think you need OS X server at all. And - that's where I think the remarks come from.
I never really understood why people who obviously don't have a clue are an ass to those that they are asking for help.