Lindbergh Beach was formerly known as Mosquito Bay.²

A photo from around 1918^2p40. Includes Saba Island and Dove Cay.

A partial view of a c. 1918 St. Thomas map showing Mosquito Bay^2p46.

Apparently the beach has been a great beach for a long time, as can be implied in the following quote from a 1918 book:

The road to the westward leads on to Mosquito Bay, where there is opportunity for excellent sea bathing^2p121