I wrestled with this exact same issue. I wandered Home Depot looking for a solution and I found it. There is a rubber seal that is designed for the bottom of overhead garage doors. it's a long strip but if you look at the end it's in the shape of a horse shoe. it's much thicker than inner tube, flexible but rigid enough that it keeps it shape fairly well.
Take it out of the box and cut off a strip that will go from end to end on your hinge. Lay it out flat and you will notice a lip on both edges. you will want to keep one edge, you'll understand why in a moment, and carefully slice off the opposite edge with a straight edge and sharp razor knife.
Place the strip over the hinge opening so that the natural curve of the strip turns toward the nest. The remaining lip should go up and against the side of the coop. The cut edge will be facing down and lay on the nest cover.
Using short exterior decking screws, attach the flap to the side of the coop, but NOT into the nest lid. If you place each screw just under the rubber lip on the top edge it will press this lip tightly against the coop.
Now run a bead of exterior black silicone seal along the entire upper lip where it meets the coop and smooth it into place with a wet finger to make a tight seal.
voila, you are done. By leaving the bottom edge free it will slide freely as you open and close the nest lid. The remaining lip makes a great semi-rigid surface to caulk.
Mine has remained flexible and dry through 105 degree days down to about 12 degrees. It's been through one monsoon style rain and not a drop of moisture in the nest.
I'l try and shoot some pics if you're interested.