Measure the diameter of the hole with a tape measure. Purchase a wooden dowel rod that is the diameter of the hole.
Push the dowel rod as far as it will go into the hole. Use a pencil to mark where the dowel is flush with the surface of the door. Remove the dowel from the door. Measure and cut the dowel so it is 1/64-inch longer than the length of the hole. If the hole goes all the way through the door, cut the dowel so that it is 1/32-inch longer that the width of the door. The dowel needs to be slightly longer than the length of the hole.
Apply some exterior epoxy to the inside of the hole. Brush the dowel rod with a thin layer of epoxy as well. Push the dowel rod into place. There should be 1/64-inch of the rod sticking out of the hole, and 1/32-inch sticking out of each side if the hole goes all the way through the door. Wipe away any excess epoxy with a rag. Allow the epoxy to dry as directed on the product instructions.
Sand the edges of the dowel down until it is flush with the surface of the door. Do not attempt to do this before the epoxy dries completely or else the dowel may move as you sand it.
Paint the dowel with paint or stain that matches the rest of the door.