In New York Harbor, about 2 miles south of the Statue of Liberty, stands a lesser-known beacon, stirring in its own way, and woven for even longer into the maritime history of New York City.

It is Robbins Reef Light Station, also known as “Kate’s Light” in honor of the pioneering woman who once tended it. Standing near the mouth of the Kill Van Kull, the cast-iron tower has safeguarded the shipping channel since 1883.

The...