Etymology
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Rob Roy (n.)

Highland freebooter and folk hero, Robert Roy MacGregor (1671-1734). His name means "Red Robert." Scott's novel first was published in 1817. As a type of cocktail made with Scotch whiskey, it is attested from 1960.

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screwdriver (n.)

also screw-driver, "tool like a blunt chisel which fits into the nick in the head of a screw and is used to turn it," 1779, from screw (n.) + driver. Meaning "cocktail made from vodka and orange juice" is recorded from 1956. (Screwed/screwy have had a sense of "drunk" since 19c.; compare slang tight "intoxicated," or perhaps the notion is "twisted").

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Bloody Mary 

the cocktail, attested from 1947 (originally touted in part as a hangover cure), said to be named for Mary Tudor, queen of England 1553-58, who earned her epithet for vigorous prosecution of Protestants. The drink earned its, apparently, simply for being red from tomato juice. The cocktail's popularity also coincided with that of the musical "South Pacific," which has a character named "Bloody Mary."

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Alexander 

masc. proper name, from Latin, from Greek Alexandros "defending men," from alexein "to ward off, keep off, turn (something) away, defend, protect" + anēr (genitive andros) "man" (from PIE root *ner- (2) "man"). The first element perhaps is related to Greek alke "protection, help, strength, power, courage," alkimos "strong;" and cognate with Sanskrit raksati "protects," Old English ealgian "to defend."

As a kind of cocktail recipe featuring crème de cacao and cream, Alexander is attested from 1913; the reason for the name is unclear.

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Martini (n.)

1891, short for Martini cocktail (1886), perhaps from Martini & Rossi, Italian firm that makes vermouth (an ingredient of the drink); the firm was in existence then by that name, but it is not specified among the ingredients in the earliest recipes (such as Harry Johnson's "Bartender's Manual," 1888). Another theory holds that it is a corruption of Martinez, California, the town where the drink was said to have originated. See discussion in Lowell Edmunds' book "Martini, Straight Up" (1998).

As the name of a type of rifle used by the British army from 1871 to 1891, it is attested from 1870, from Friedrich von Martini, who invented the breech mechanism on it.

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mojito (n.)

type of rum-based Cuban cocktail, by 1946, from Cuban Spanish, a diminutive of mojo, a word for certain sauces and marinades; Ayto ("Diner's Dictionary") considers it  to be "probably a reapplication of the Spanish adjective mojo 'wet,'" from mojar "to moisten, make wet," from Vulgar Latin *molliare"to soften by soaking," from Latin mollire "to soften" (see emollient). 

MOJITO
I don't know who originated this one, but every bar in the West Indies serves it, practically every rum recipe booklet gives the formula for it, so my little collection of rum drinks would hardly be complete without it. Such popularity must be deserved, and it is. It's a swell drink! ["Trader Vic's Book of Food and Drink," 1946]
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cosmopolitan (adj.)

1815, "free from local, provincial, or national prejudices and attachments," from cosmopolite "citizen of the world" (q.v.) on model of metropolitan. From 1833 as "belonging to all parts of the world, limited to no place or society." Meaning "composed of people of all nations, multi-ethnic" is from 1840. The U.S. women's magazine of the same name was first published in 1886.

As a noun, "one who is at home all over the world, a cosmopolite," 1640s. As the name of a vodka-based cocktail popular in 1990s (due to "Sex and the City" TV program) from late 1980s (the drink itself seems to date to the 1970s).

Cosmopolitanism in reference to an ideology that considers all humans as a single community is recorded by 1828. It took on a negative tinge in mid-20c., suggesting an undermining of indigenous and national societies and often tied to the supposed influence of the Jews.

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gimlet (n.)

type of boring tool, mid-14c., gymbelette, from Anglo-French and Old French guimbelet, guibelet (12c., Modern French gibelet), which is probably of Germanic origin, perhaps from Middle Dutch wimmelkijn (with substitute of French diminutive suffix), diminutive of wimmel "auger, drill," which is perhaps from a nasalized form of PIE root *weip- "to turn" on the notion of "That which turns in boring." Middle English also had wimble in the same sense (mid-13c.), probably from an Old North French form of the same word.

As the name of a cocktail made with gin or vodka and (Rose's) lime juice, by 1927, apparently originally nautical, presumably from its "penetrating" effects on the drinker (a gimlet was the tool used to tap casks). There also was a British Navy surgeon named Gimlette at the turn of the 20th century who was active in health matters. Popularized in the U.S. during prohibition as being quick and easy to mix, and the lime masked the scent.

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Gibson girl (n.)

"woman considered stylish at the turn of the 20th century," 1894, named for U.S. artist and illustrator Charles Dana Gibson (1867-1944), whose main model was his wife, Irene Langhorne (1873-1956). The Gibson cocktail (gin, vermouth, and a pearl onion) is attested by 1914, in some stories ascribed to him but the origin of the term is unknown.

"She looks like a Gibson girl" is not an uncommon saying; and to look like a Gibson girl, is not without its merits. Although our artist has expressed in his drawings disapproval of women usurping the spheres of men, his girls suggest intellectuality. He has none of the doll-like inanely pretty faces which artists used to give women in olden days. His girls look as if they would have opinions of their own and would act with discrimination in the affairs of life. They are tall and graceful and although not in the least like fashion plates, their clothes are becoming and fit perfectly. [National Magazine, May 1898]
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prairie-oyster (n.)

"salted and peppered raw egg, drunk in booze or vinegar," by 1878, American English, from prairie + oyster (in reference to the taste or the method of consuming it). Also called prairie-cocktail (1889). Prairie-oyster as "fried calf testicle," considered a delicacy, is by 1941.

PRAIRIE OYSTER. This simple but very nutritious drink may be taken by any person of the most delicate digestion, and has become one of the most popular delicacies since its introduction by me at Messrs. Spiers and Pond's. Its mode of preparation is very simple. Into a wine glass pat a new-laid egg ; add half a tea-spoonful of vinegar, dropping it gently down on the inside of the glass ; then drop on the yolk a little common salt, sufficient not to quite cover half the size of a threepenny-piece; pepper according to taste, The way to take this should be by placing the glass with the vinegar furthest from the mouth and swallow the contents. The vinegar being the last gives it more of an oyster-like flavour. [Leo Engel, "American & Other Drinks," London, 1878]
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