My wife and I joined a turtle-watch sponsored by the park service. We met in the nearby parking lot at dark and walked out to the beach where we hung out until about 12:30 when we packed it in without seeing any turtles. The purpose of the activity is to watch egg-laying and then relocate the eggs inland where they are safer. The area is patrolled by the park police to prevent thefts of the eggs by people and dogs.
Although we did not see any turtles during this visit, it was well worth the time. An extremely knowledgeable young park ranger talked non-stop for about 3 hours about turtles, the environment in the area, and anything else we asked. Hats off to her because we learned a lot, she was super interesting, and everyone was enthusiastic. If you have younger kids, they will also enjoy this free lesson.
AND, we got to see the Southern Cross constellation - first time for me!
If it matters, this is the beach used in the final scenes of the Shawshank Redemption movie.
I'm not sure how to arrange a trip as some friends who are residents on the island set it up, but you could call the park service and ask.