polyurethane is one of the most common wood finishes on marine applications. the unfortunate thing is that sunlight will damage it in about a year. on polished surfaces it will start to chip off pretty fast. on larger pieces i've found lacquer to last longer.
on small parts that need wear and moisture resistance i use CA glue. it dries clear, sticks great, stabilizes cracks and splinters and polishes up to a high gloss without too much effort. depending on your string pressure, material type and use you may have to refinish it every few months.
thinned, boiled linseed and a some other types of oils may also be a good option since it's on a working part. they will rub in instead of off.
if you have any left over pieces of the bone you are using, you may want to experiment with different ideas first. some permanent markers, fruit juices, stains, oils, etc may penetrate more than others. some finishes may also adhere better. this is one of the times where you have to figure out what the material will allow you to accomplish. bone and antler from one animal can have drastically different properties.
if you look at the "skullarina" pic you'll see that along the cheek arches where my hands rub, the lacquer has worn off and the oils from my hands have stained it darker than the rest. you could just leave it unfinished for a while and rub some sweaty, not so clean hands on it regularly. to me the best patina is the one you create with time and use.