I don't have any data, but surely it depends on the application.
Wikipedia, being an encyclopaedia, has a "user mode" of "browsing" (if you like) where a user will be using the site in an effectively random mode just following links that take their fancy. In this mode then a "show me a random page" function has real utility.
If however, I was using the site to find some specific piece of information then such a function would be useless and were I to actually click it counter productive.
So if your application has this "browsing" mode then a random function might be useful. I think you'd need to get some usage data to see if that's a valid model.
As an aside Stack Exchange might benefit from such a "Random Question" function to show a random unanswered question. In fact it has already been suggested here