The Daiquiri is one of six basic drinks listed in David A. Embury's classic The Fine Art of Mixing Drinks. In the book, he also suggests some variations, such as substitute part or all of syrup with grenadine.
Originally the drink was served in a tall glass packed with cracked ice. A teaspoon of sugar was poured over the ice and the juice of one or two limes was squeezed over the sugar. Two or three ounces of rum completed the mixture. The glass was then frosted by stirring with a long-handled spoon. Later the Daiquiri evolved to be mixed in a shaker with the same ingredients but with shaved ice. After a thorough shaking, it was poured into a chilled flute glass. An article in the March 14, 1937 edition of the Miami Herald as well as private correspondence of J.F. Linthicum confirm the recipe and early history.
Consumption of the drink remained localized until 1909, when Admiral Lucius W. Johnson, a U.S. Navy medical officer, tried Cox's drink. Johnson subsequently introduced it to the Army and Navy Club in Washington, D.C., and drinkers of the daiquiri increased over the space of a few decades. The daiquiri was one of the favorite drinks of writer Ernest Hemingway and president John F. Kennedy.
The drink became incredibly popular in the 1940s. Wartime rationing made whiskey, vodka, etc, hard to come by, yet because of Roosevelt's Good Neighbor policy (which opened up trade and travel relations with Latin America, Cuba and the Caribbean), rum became highly attainable. The Good Neighbor Policy (also known as 'The Pan-American program'), helped make Latin America seem fashionable. As a consequence, rum-based drinks (once frowned upon as being the domain of sailors and down-and-outs), also became fashionable, and the Daiquiri saw a tremendous rise in popularity in the US.
The basic recipe for a Daiquiri is also extremely similar to what British sailors drank aboard ship from the 1740s onwards - Grog - which is basically rum mixed with sweetened preserved lime juice and water. This was a common drink across the Caribbean, and as soon as ice became available this was included instead of the water. Jennings-Cox's story is certainly a popular one & maybe he was responsible for the naming of the drink, but as far as creating it he was about 150 years late.