I have narcolepsy, which might preclude me from any normal research into sleep-Internet use.
I write email and forum posts in my sleep on a semi-recurring basis [the more stressed out I am and physically exhausted, the more often it occurs].
It is really interesting [and kind of terrifying, the "oh dear, what did I write this time, and who to?" sinking feeling of realization when I wake up to different browser page open than expected] to see the fluidity of the words I write. More free-flowing, less choppy, but still recognizably my style of writing and thoughts.
I think it would be really interesting if there was a way that an ongoing polysomnography could be preformed, to correlate with what I post/write and when. The sleep lab that my doctor uses does not allow computers to be on when testing, lights out at 10-11pm, and up at 7am -- which does not at all represent my typical sleep behavior of falling asleep somewhere around 2am, getting up between 10-12noon, generally curled around laptop with password-requiring screensaver.
Passwords really aren't that hard to type in sleep - it is patterns that you memorize of how your hands move, when you type them in enough.
The interesting part is that she had her hands aligned on keyboard properly to be hitting correct keys in sleep -- some of the stuff I've typed in sleep looks like gibberish, as the nubs on my F and J keys are worn down, so I guess I've ended up with hands slightly over from where they should be.