Good question. We didn't know the answer, either, so we did what anyone does these days as a first pass and googled it. This brought up two conflicting theories about the nickname of the man whose own official Web site obituary describes him as, "the Las Vegas gangster and Black Book member played by Robert DeNiro in the movie Casino," but does not mention when, where, or how he got the nickname "Lefty."
According to the Wikipedia entry for Frank Rosenthal, he was nicknamed "Lefty" because, "during a Congressional subcommittee hearing, he pleaded the Fifth Amendment 37 times, not even answering the question of whether or not he was left-handed." It goes on to say, "In fact, however, he had the nickname from childhood, simply because he was left-handed."
Both these explanations were repeated in various places and neither of them had the ring of truth, so we put in a call to Las Vegas Review-Journal columnist and general expert on the city's mob past, John L. Smith. John's version, which he was told separately by Marty Kane and Joey Boston, themselves both Black Book members who'd worked as bookies for Rosenthal (and are now also deceased), was that during a raid on a bookmaking operation in Miami, Rosenthal cut a tendon in his right hand/arm while smashing a window and, hence, was forced to use his left hand for a while, which earned him the nickname Lefty.
This is the most plausible-sounding account we've heard and it's the one we'll stick with, until such time as someone can come up with a verifiable alternative.
Our thanks to John, who in addition to writing for the R-J is also the author of several Huntington Press titles, including Of Rats and Men (a biography of Mayor Oscar Goodman), Quicksilver: The Ted Binion Murder Case, No Limit: The Rise and Fall of Bob Stupak and the Stratosphere Tower, and On the Boulevard: The Best of John L. Smith.