The origin of the word “cocktail” (alcoholic mixed drink) remains a mystery. One early theory supposed that it was served by Elizabeth Flanagan in Westchester County, around the time of the Revolutionary War. Early citations for “cocktail” do seem to come from New York.
 
 
Wikipedia: Cocktail
A cocktail is a style of mixed drink. However, not all mixed drinks are cocktails. A cocktail usually contains one or more types of liquor and flavorings and one or more liqueurs, fruit juices, sauces, honey, milk, cream or spices, etc.
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History
The earliest known printed use of the word “cocktail,” as originally determined by David Wondrich in October 2005, was from “The Farmer’s Cabinet”, April 28, 1803, p [2]: “11. Drank a glass of cocktail — excellent for the head ... Call’d at the Doct’s. found Burnham — he looked very wise — drank another glass of cocktail.”
 
The second earliest and officially recognised known printed use of the word “cocktail” (and the most well-known) was in the May 13, 1806 edition of the Balance and Columbian Repository, a publication in Hudson, New York , where the paper provided the following answer to what a cocktail was:
 
“Cocktail is a stimulating liquor composed of spirits of any kind, sugar, water, and bitters — it is vulgarly called a bittered sling and is supposed to be an excellent electioneering potion, inasmuch as it renders the heart stout and bold, at the same time that it fuddles the head. It is said, also to be of great use to a Democratic candidate: because a person, having swallowed a glass of it, is ready to swallow anything else.”
 
It is believed that the term “cocktail” was first used in the village of Elmsford in Westchester County, New York after a local bar ran out of stirrers and resorted to use a cock’s tail feathers to stir the drink.
   
(Oxford English Dictionary) 
cocktail, n. and adj.
[A slang name, of which the real origin appears to be lost.]
A drink, consisting of spirit mixed with a small quantity of bitters, some sugar, etc. orig. U.S.
1803 Farmer’s Cabinet 28 Apr. 2/3 Drank a glass of cocktail—excellent for the head… Call’d at the Doct’s…drank another glass of cocktail.
1806 Balance (Hudson, N.Y.) 13 May 146 Cock tail, then, is a stimulating liquor, composed of spirits of any kind, sugar, water, and bitters.
1809 ‘D. KNICKERBOCKER’ Hist. N.Y. (1861) 241 They lay claim to be the first inventors of those recondite beverages, cock-tail, stone-fence, and sherry-cobbler.
 
1789? The Prelateiad; or, the Rape of the Holy Bottle 22 (Eighteenth Century Collections Online):
All Ceylon’s spicy gifts its moisture mends, And Kyan’s Pep. its cock-tail virtue lends.
 
13 May 1806, Balance (Hudson, NY), pg. 146:
Cock-tail, then, is a stimulating liquor, composed of spirits of any kind, sugar, water., and bitters—it is vulgarly caled bittered sling, and is supposed to be an excellent electioneering potion, inasmuch as it renders the heart stout and bold, at the same time that it fuddles the head. It is said also, to be of great use to a democratic candidate: because a person having swallowed a glass of it, is ready to swallow any thing else.
Edit. Bal.
 
24 May 1806, Pittsfield (MA) Sun, pg. 4:
Is it by the honest means of argument or sound reasoning? Or is it by publishing grog stories, and strictures on “cock tail”?
 
The Spy
by James Fenimore Cooper
New-York, NY: Wiley & Halsted
1821
Volume 1, Chapter XVI
Pg. 236:
On a rough board suspended from the gallows looking post that had supported the ancient sign was, however, written in red chalk “Elizabeth Flanagan, her hotel,” an ebullition of wit from some of the idle wags of the corps. The matron, whose name had thus been exalted to an office of such unexpected dignity, ordinarily discharged the duties of a female sutler, washerwoman, and, to use the language of Katy Haynes, bitch-doctor to the troops; she was the widow of a soldier who had been killed in the service, and who, like herself, was a native of a distant island, that had early tried
his fortune in the colonies of North America. She constantly migrated with the troops, and it was seldom that they became stationary for two days at a time, (Pg. 237—ed.) but the little cart of the bustling woman was seen driving into their encampment, loaded with such articles, as she conceived would make her presence most welcome. With a celerity that seemed almost supernatural, Betty took up her ground and commenced her occupation; sometimes the cart itself was her shop; at others, the soldiers made her a rude shelter of such materials as offered; but on the present occasion she had seized on a vacant building, and by dint of stuffing the dirty breeches and half dried linen of the troopers in the broken windows, to exclude the cold which had now become severe, she formed what she herself had pronounced to be “most iligant lodgings.”
 
The men were quartered in the adjacent barns, and the officers collected in the “Hotel Flanagan,” which they facetiously called headquarters. Betty was well known to every trooper in the corps, could call each by his christian or nick-name, as best suited her fancy; and, although absolutely intolerable to all whom habit had not made familiar with her virtues, was a general favorite with these partizan warriors. Her faults were, a trifling love of liquor, excessive filthiness, and a total disregard to all decencies of language; her virtues, an unbounded love for her adopted country, perfect honesty when dealing on certain known principles with the soldiery, and great good nature: added to these, Betty had the merit of being the inventor of that beverage which is so well known at the present hour, to all the patriots who make a winter’s march between the commercial and political capitals of this great state, and which is distinguished by the name of “cock-tail.” Elizabeth Flanagan was peculiarly well qualified by education and circumstances to perfect this improvement in liquors, having been literally brought up on its principal ingredient, (Pg. 238—ed.) and having acquired from her Virginia customers the use of mint, from its flavour in a julep, to its height of renown in the article in question. Such, then, was the mistress of the mansion…
   
The Ultimate Cocktail Book II
by Raymond Foley
Foley Publishing Corporation
1998
Pp. 64-67 (origin of the cocktail, with a passage from the book below):
The Cocktail Book
by R. L. Paget (pseudonym)
new revised edition
Boston, MA: L. C. Page & Company
1913
   
In a famous old tavern not far from the Philipse Manor House, the site of what is now Yonkers on the Hudson, and the very centre of the most popular sport of the times, was blended the first delightful cocktail. If the descendants of William Van Eyck, its jolly host, may be believed…  (From his champion game-cock—ed.) Thus was the drink named. And, in after days, when Master Appleton kept the tavern, its sign was the sign of the Cock’s Tail, which ever proved an emblem of good fortune to him and his good wife, their children and their children’s children.