According to A.S Crockett in The Old Waldorf-Astoria Bar Book, the National Cocktail was a signature drink of the Hotel National in Havana, Cuba. Crockett was introduced to it by Will P. Taylor who managed the National and, at one time, the Old Waldorf-Astoria. At the hotel this drink was also ordered as a "Daiquiri, or a Bacardi" though it is far different from those two cocktails.
Some newer variations of the National Cocktail cut the pineapple to a quarter of the rum and use lime instead of lemon. Personally I prefer this "old-school" version.