Etymology
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gin (v.1)

in slang phrase gin up "enliven, make more exciting," 1887 (ginning is from 1825), perhaps a special use of the verb associated with gin (n.2) "engine," but perhaps rather or also from ginger up in the same sense (1849), which is from ginger in sense of "spice, pizzazz;" specifically in reference to the treatment described in the 1796 edition of Grose's slang dictionary under the entry for feague:

... to put ginger up a horse's fundament, and formerly, as it is said, a live eel, to make him lively and carry his tail well; it is said, a forfeit is incurred by any horse-dealer's servant, who shall shew a horse without first feaguing him. Feague is used, figuratively, for encouraging or spiriting one up.
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gin (v.2)
"to begin," c. 1200, ginnen, shortened form of beginnen (see begin).
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gin (n.1)

type of distilled drinking alcohol, 1714, shortening of geneva, altered (by influence of the name of the Swiss city, with which it has no connection) from Dutch genever "gin," literally "juniper" (because the alcohol was flavored with its berries), from Old French genevre "the plant juniper" (12c.), from Vulgar Latin *jeniperus, from Latin juniperus "juniper" (see juniper).

[I]t was not till about 1724 that the passion for gin-drinking appears to have infected the masses of the population, and it spread with the rapidity and the violence of an epidemic. Small as is the place which this fact occupies in English history, it was probably, if we consider all the consequences that have flowed from it, the most momentous in that of the eighteenth century—incomparably more so than any event in the purely political or military annals of the country. [W.E.H. Lecky, "A History of England in the Eighteenth Century," 1878]

Gin and tonic is attested by 1873; gin-sling by 1790; gin-fizz (with lemon juice and aerated water) is from 1878. Gin-mill, U.S. slang for "low-class tavern or saloon where spirits are drunk" (1872) might be a play on the senses from gin (n.2). British gin-palace "gaudily decorated tavern or saloon where spirits are drunk" is from 1831.

The card game gin rummy first attested 1941 (described in "Life" that year as the latest Hollywood fad); OED lists it with the entries for the liquor, but the sense connection seems obscure other than as a play on rummy.

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gin (n.2)
"machine for separating cotton from seeds," 1796, American English, used earlier of other machineries, especially of war or torture, from Middle English gin "ingenious device, contrivance" (c. 1200), from Old French gin "machine, device, scheme," shortened form of engin (see engine). The verb in this sense is recorded from 1789. Related: Ginned; ginning. Middle English had ginful "ingenious, crafty; guileful, treacherous" (c. 1300).
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geneva (n.)
1706, alteration (by influence of the Swiss city name) of Dutch genevre, French genière (see gin (n.1)).
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Jenny 

fem. personal name, originally another form of Jane, Janey and a diminutive of Jane or Janet; in modern use (mid-20c.) typically a shortening of Jennifer. Jenny is attested from c. 1600 as female equivalent of jack (n.), and like it applied to animals (especially of birds, of a heron, a jay, but especially Jenny wren, 1640s, in bird-fables the consort of Robin Redbreast). Also like jack used of machinery; Akrwright's spinning jenny (1783) is said to have been named for his wife, but is perhaps rather a corruption of gin (n.2) "engine."

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cotton (n.)

late 13c., "white fibrous substance containing the seeds of the cotton plant," from Old French coton (12c.), ultimately (via Provenal, Italian, or Old Spanish) from Arabic qutn, a word perhaps of Egyptian origin. Also ultimately from the Arabic word are Dutch katoen, German Kattun, Provenal coton, Italian cotone, Spanish algodon, Portuguese algodo.

As "cloth made of cotton," from early 15c. Meaning "the cotton plant" is from c. 1400. As an adjective, "made of cotton," from 1550s. Cotton gin is recorded from 1794 (see gin (n.2)). Philip Miller of the Chelsea Physic Garden sent the first cotton seeds to American colony of Georgia in 1732.

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ginger (n.)

11c., from Old English ginȝifer, ginȝiber, from Late Latin gingiber, from Latin zingiberi, from Greek zingiberis, from Prakrit (Middle Indic) singabera, from Sanskrit srngaveram, from srngam "horn" + vera- "body," so called from the shape of its root. But this may be Sanskrit folk etymology, and the word may be from an ancient Dravidian word that also produced the modern name for the spice, inchi-ver (inchi "ginger", ver "root"). 

Bishop Caldwell and Drs. Burnell and Gundert considered that the Tamil iñci must have had an initial ś- formerly, that the Sanskrit śṛṅgabera was an imitation of the (supposititious) Tamil ciñcivēr and that European zingiber was derived from the Tamil name. [R. Swaminatha Aiyar, Dravidian Theories]

The word apparently was readopted in Middle English from Old French gingibre (12c., Modern French gingembre). In reference to coloring, by 1785 of fighting cocks, 1885 of persons (gingery with reference to hair is from 1852). The meaning "spirit, spunk, temper" is from 1843, American English (see gin (v.1)).

Ginger-ale is recorded by 1822, the term adopted by manufacturers to distinguish their product from ginger beer (1809), which was sometimes fermented. Ginger-snap as a type of hard cookie flavored with ginger is by 1855, American English.

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old-fashioned (adj.)

1650s, "in an outdated style, formed in a fashion that has become obsolete," from old + past participle of fashion (v.). Meaning "partaking of the old ways, suited to the tastes of former times" is from 1680s. Related: Old-fashionedness. New-fashioned is recorded from 1610s.

As a type of cocktail, Old Fashioned is attested by 1901, American English, short for a fuller name.

Old Fashioned Tom Gin Cocktail Mix same as Holland Gin Old Fashioned Cocktail using Old Tom gin in place of Holland [George J. Kappeler, "Modern American Drinks," Akron, Ohio, 1900]

(Old Tom (1821) was a name for a strong variety of English gin.)

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bathtub (n.)

also bath-tub, "a tub to bathe in," especially as a permanent fixture of a bathroom, 1837, from bath + tub. Prohibition-era bathtub gin is recorded by 1928.

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