We've now got one of those collapsible cloth wagons and the scorebook fits under the med kit, the "drying ball off" beach towels, and anything else we can pile on when we are caught out in the rain. Of course, if you are still playing and not in a covered dugout when it starts to rain, there isn't a heck of a lot you can do to protect the scorebook.
But I really do recommend those wagons. An unanticipated benefit is that the girls love to volunteer to pull the wagon.