In 1902 workers completed a mysterious tower, 187 feet high and shaped like a giant mushroom, on which rested the hopes of one of the 20th century’s most prolific geniuses.

Facing the beach in the hamlet of Shoreham, N.Y., on Long Island, the Wardenclyffe Tower was, according to its inventor, Nikola Tesla, the key that could unlock an age of wonders.

As...