I'm voting that the stunned cannot do anything for one entire round, and so skipps the next available action. Now, it's quite possible that *after* that next action the stunned could get a new initiative roll if you don't roll every round.
If you do roll every round, then you could have the stunned roll, and do nothing for that action, and roll again next. If you don't roll, the stunned does nothing on his "turn", then either acts again after missing a turn, or potentially rolls initiative again. But that's as far as I'd go towards letting the character potentially act faster than normal (the allowing a possible better initiative roll, as long as you keep that, it's also having a potenitally worse one.