gin (n.1) Look up gin at Dictionary.com
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). 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.
gin (n.2) Look up gin at Dictionary.com
"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).
gin (v.1) Look up gin at Dictionary.com
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 1811 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.
gin (v.2) Look up gin at Dictionary.com
"to begin," c.1200, ginnen, shortened form of beginnen (see begin).