I would put some kind of flashing on the crack where the hinges are and then caulk the heck out of where it attaches to the wall so that no water could get behind it. Or run the flashing all the way up to underneath the roofline.
And then maybe put shingles on the lid, diverting water away and down. I can't really see if there is a slope to the lid, but if not, I would seriously consider redesigning the box construction so that water flows down and can't backup into the coop.