A sysadmin use for Twitter
We have some users that are interested in reading about technical system status updates and what the sysadmins are doing in general. The obvious solution is some sort of blog-like environment where sysadmins write things periodically, but the problem with an actual blog is that it takes too much time to write something, especially if we are in the middle of a semi-crisis.
Hence the attraction of Twitter. The short length of Twitter messages (and their lack of formatting) means that we simply can't write very much and in turn users can't demand very much (and can't get disappointed when we fail to turn out polished marvels of educational clarity). Twitter is also well supplied with clients, including command line clients, that are basically fire and forget; you type your message at a command line or into a text box, and you're done.
(I have seen Twitter described as micro-blogging, which is just what we want; something like a blog but much smaller and easier to deal with, and with lower social expectations from the people reading it.)
You could use a Twitter clone for this and host it yourself, but for this specific purpose I think you might as well use Twitter itself. Among other reasons, I suspect that many of the users who would be interested in this are the sort of people who already have Twitter accounts. (And if not, Twitter has syndication feeds.)