Cognitive sychologists study this kind of question, as you might expect. Here's a paper (behind a paywall, sorry) where they asked people to name random digits. You don't get uniform distribution on 0,..,9.
I learned a little about this stuff when I was writing a blog post about detecting election fraud by looking for digits which looked more like "numbers made up by humans" than "numbers arrived at randomly."
Update: I spoke to my colleague Gary Lupyan, a cognitive psychologist here who studies such things. There are lots of interesting results, although he hasn't done the precise experiment suggested in the question. If you ask people to name a number between 1 and 100, the modal responses are between 1 and 10, with maybe a slight continuing dropoff afterwards. People disprefer even numbers and multiples of 5 and 10. He also replicated the folk belief that if you ask people to name a number between 1 and 20, the modal response is 17.
It doesn't look to me like the results he's getting are well-modeled by any particularly natural distribution, though you could certainly fit some kind of decay to it.