Place a board underneath the damaged area of the rug. Cut out the damaged spot in a square, being careful to cut through the backing attached to the hide.
Set the square you've cut on top of a matching piece of cowhide. Hides have a grain, or direction that the hairs run in. Check to be sure the hairs on the patch piece of hide are running the same direction as the damaged piece. This will help your patch last and render it almost invisible.
Lay the new piece on a length of carpet backing. Cut the backing around the square, leaving a 1-inch border of backing along each edge of the square.
Apply fabric adhesive to the back of the patch. Press it firmly onto the backing, and allow it an hour or two to dry completely.
Apply fabric adhesive to the top side of the backing fabric border. Place the patch underneath the cut-out area and adjust it to fit. Weight the top of the carpet over the patch until it's dry.
Turn the cowhide rug over and stitch the edges of the backing to the carpet with a curved darning needle and thread. Be sure to use a thimble on the finger you use to push the needle through the tough fabric.