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.