depends on which information to believe
George Sinclair's 2007 article on the origin of the drink quotes the New York Herald Tribune from 1948:
The mule was born in Manhattan but "stalled" on the West Coast for the duration. The birthplace of "Little Moscow" was in New York's Chatham Hotel. That was back in 1941 when the first carload of Jack Morgan's Cock 'n' Bull ginger beer was railing over the plains to give New Yorkers a happy surprise… The Violette Family helped. Three friends were in the Chatham bar, one John A. Morgan, known as Jack, president of Cock 'n' Bull Products and owner of the Hollywood Cock 'n' Bull Restaurant; one was John G. Martin, president of G.F. Heublein Brothers Inc. of Hartford, Conn., and the third was Rudolph Kunett, president of the Pierre Smirnoff, Heublein's vodka division. As Jack Morgan tells it, "We three were quaffing a slug, nibbling an hors d'oeuvre and shoving toward inventive genius". Martin and Kunett had their minds on their vodka and wondered what would happen if a two-ounce shot joined with Morgan's ginger beer and the squeeze of a lemon. Ice was ordered, lemons procured, mugs ushered in and the concoction put together. Cups were raised, the men counted five and down went the first taste. It was good. It lifted the spirit to adventure. Four or five days later the mixture was christened the Moscow mule...
According to a 1942 Insider Hollywood article, the Moscow mule was most popular in Los Angeles, where it originated. The Nevada State Journal (12 October 1943) reinforced the mule's popularity in reporting: "Already the mule is climbing up into the exclusive handful of most-popular mixed drinks". It became known as a favorite drink of Reno casino owner William F. Harrah. In his book Beat the Dealer (1964), Edward O. Thorp did not name the Tahoe casino where he thought he had been poorly treated as a card counter. Instead, he wrote, "Immediately I had a Moscow mule", subtly hinting that the location was Harrah's Lake Tahoe, due to Harrah's then well-known proclivity for the drink.