Lady Franklin Bay is named for a Victorian woman Lady Jane Griffen Franklin, wife of famous British explorer Sir John Franklin who vanished from Baffin Bay beyond Lancaster Sound on the HMS Erebus in 1845 while attempting to trace the Northwest Passage.
Originally named "Mount Arthur Eugene" in 1883 by the States Army Signal Service during their Lady Franklin Bay expedition.
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The strait is named after Sergeant George W. Rice (born 29 June 1855 in Baddeck, Nova Scotia), who was the photographer on Adolphus Greely's ill-fated Lady Franklin Bay Expedition, and also a correspondent with the New York Herald.
One of the duties of the U.S. Army Signal Corps at the time was the management of the Weather Service and Hazen criticized the government's lack of response to the distress of the International Polar Year expedition to Fort Conger, Lady Franklin Bay (on Ellesmere Island, Canada).