Looking at where this hole is positioned and the damage around the edge I would hazard to say that this was caused by an insect chewing through from the other side. I have seen this same damage on other horn hilts before where the offending insect had left a thin layer of horn on one end that had broken away later leaving the same raged edge shown here. Is there a hole on the other side of the finger and if so does it look straight, round, smooth on the inside and the same diameter all the way through like it was made by a drill or is the hole a bit larger inside the finger than where it first starts? If it is larger on the inside then it is bug damage, if not I am at a loss. There is no reason that I can think of to drill a hole of this size into the side of the finger as this one is to mount a ring or anything else I can think of.
If I were doing a restoration on this piece I would suggest to its owner (if he wanted this repaired) that the hole on the side that shows be evened up to remove the chipped edges. If this hole goes completely through the finger as I believe is does I would close the other end using something that could be easily removed latter like a piece of masking tape. Then fill the void about 3/4's of the way with a mixture of epoxy and black paint pigment stirring it around the inside with a toothpick or other appropriate item to insure the epoxy has a firm attachment to the entire inside surface of the hole. I would then carve a small piece of horn to create a tightly fitting plug with the grain and color matching as closely as possible to the rest of the hilt. Using the same epoxy pigment mixture fit this piece into the hilt using enough pressure to ensure as close to an invisible seam as possible. Clamp or tie the plug to maintain pressure until the epoxy dries. Regardless of what the adhesives instructions tell you let this to dry for a minimum of 48 hours. Once dry you can carve away the excess material to match the surrounding area and polish to match. Photo below of similar damage.