I am pretty sure it has always had the same name.
Longacre Theatre - 1564 Broadway @ 46th Street. Built in 1913 as the Longacre Theatre. Notable Productions include Leave it to Jane (1917), The Butter and Egg Man (1925), Waiting for Lefty (1935), Mark Twain Tonight (1966), The Me Nobody Knows (1971), The Ritz (1975), Ain't Misbehavin' (197

, Harrigan 'n Hart (1985), Russell Simmons Def Poetry Jam (2002).