The 1876 Centennial Exhibition was much like the World's Fairs of the mid 20th century. The HL382 actually commemorates two separate exhibits.
There was much hype and excitement being generated by the press about France's gift of the Statue of Liberty. Although the sculpture wasn't quite completed yet, the torch was packed up and rushed to the states in order to be displayed at the Exposition. Patrons were charged a fee to climb to the top to view the entire Fairgrounds. Money raised from this extremely popular climb helped build the base for the statue.
As for the lighthouse, the Lighthouse Board was in the process of building the Ship John Shoal Lighthouse to be erected in New Jersey waters. It is that lighthouse which was on display at the Centennial Exposition. At the conclusion of the fair the lighthouse was dismantled then reassembled in its current location.