The origin of the name, Red Rover, and its usage in association with this game is not clear. Red Rover appears to have been used as a name for boats in the 18th and 19th centuries. In 1828 a steamboat called 'The Red Rover' was the first to service the Hatchie River.
In 1855, J. Fenimore Cooper, wrote a book called 'The Red Rover':
"...the lawless vagaries of the Red Rover himself, the pirate chief who glories in chaos and who orders his men 'All hands to mischief, ahoy!'"
On Christmas Eve 1862, Sisters of the Holy Cross, pioneers of the US Navy Nurse Corps, boarded the first hospital ship, The Red Rover, which aided the wounded soldiers of both sides during the American Civil War.
It is possible the name of the game derives either from the shuttling motion of a ship across a river, or, in the case of the nurses, from aid being given to both sides of a conflict.