Up, on-the-rocks or frozen
In my formative Daiquiri drinking years, I followed the convention that a Daiquiri No. 1 should be served 'straight-up'. However, I now tend to drink my Daiquiris 'on-the-rocks', and interestingly Jennings Cox's original recipe suggests that this may also be the way he originally intended the drink to be served.
In his diary, Cox stipulates, "Put all ingredients in a cocktail shaker and shake well. Do not strain as the glass may be served with some ice." And as Albert S. Crockett notes of the Daiquiri in his 1935 The Old Waldorf-Astoria Bar Book, "Personal preference dictates serving the cocktail with finely shaved ice in the glass."
Obviously, serving a drink over ice will add dilution (so rendering the addition of a splash of water to the recipe superfluous). Having tried straight-up Daiquiris and on-the-rocks Daiquiris made to the same formula next to each other, I have to admit more nuances are found in the 'up' version compared to on-the-rocks. Perhaps, like me, vary how you take your Daiquiri according to mood and weather.
There's a potentially brain-numbing third way: blended and served 'frozen'. The blended Frozen Daiquiri is said to have first been produced by Emilio Gonzalez at the Plaza Hotel in Cuba. However, it was made famous by Constantino (Constante) Ribalagua Vert who presided over the bar at Havana's La Florida (later renamed Floridita to distinguish it from the restaurant of the same name) for some forty years until his death in early December 1952.
In his 1948 The Fine Art of Mixing Drinks, David A. Embury writes of Havana's Floridita, "This restaurant, at the corner of Obispo and Monserrate streets in Havana, became known as 'La Catedral del Daiquiri' (The Temple of the Daiquiri) and Ribalagua as the Cocktail King - 'El Rey de los Coteleros'". The title was, indeed, well deserved. His limes were gently squeezed with his fingers lest even a drop of the bitter oil from the peel got into the drink; the cocktails were mixed (but not overmixed) in a Waring Blender; the stinging cold drink was strained through a fine sieve into the glass so that not one tiny piece of the ice remained in it. No smallest detail was overlooked in achieving the flawless perfection of the drink."
Ernest Hemingway, the hard-drinking, Nobel prize-winning author, lived in Cuba for years, indulging his passions for fishing, shooting and boozing. In the 30s and the 40s he would often work his way through twelve of the Floridita's frozen Daiquiris - often doubles, renamed 'Papa Dobles' in his honour. The Hemingway Special Daiquiri, which includes grapefruit, was created for him.
In his book Islands in the Stream, Hemingway's hero stares deep into his frozen Daiquiri, and Hemingway writes, "It reminded him of the sea. The frappéd part of the drink was like the wake of a ship and the clear part was the way the water looked when the bow cut it when you were in shallow water over marl bottom. That was almost the exact colour."
There's nothing like a Frozen Daiquiri to quench your thirst on a hot summer's day but why upset the peace with the whirl of an electric blender when you can have a Daiquiri Frappé. Daiquiri served frappé, shaken and served over crushed ice has all the cooling qualities of a Frozen Daquiri but without the risk of brain freeze.