rummy (n.1)

"liquor distilled from the juice of sugar cane or molasses," 1650s, apparently a shortening of rumbullion (1651), rombostion (1652), words all of uncertain origin, but suspicion falls on rum (adj.) "excellent, fine, good, valuable;" the phrase rum bouse "good liquor" is attested from 1560s and through 17c. The English word was borrowed into Dutch, German, Swedish, Danish, Spanish, Portuguese, Italian, French, and Russian.

In the Library of Trinity College, Dublin, is a manuscript entitled "A briefe description of the Island of Barbados." It is undated but from internal evidence it must have been written about the year 1651. In describing the various drinks in vogue in Barbados, the writer says : "The chief fudling they make in the Island is Rumbullion alias Kill-Divill, and this is made of sugar canes distilled, a hot, hellish, and terrible liquor. ["The Etymology of the Word Rum,"  in Timehri, 1885]

Rum was used from c.1800 in North America as a general (hostile) name for intoxicating liquors, hence rum-runner and much other Prohibition-era slang.

Rum I take to be the name which unwashed moralists apply alike to the product distilled from molasses and the noblest juices of the vineyard. Burgundy in "all its sunset glow" is rum. Champagne, soul of "the foaming grape of Eastern France," is rum. ... Sir, I repudiate the loathsome vulgarism as an insult to the first miracle wrought by the Founder of our religion! [Oliver Wendell Holmes, "The Autocrat of the Breakfast-Table," 1871]

suffix in pet proper names (such as Johnny, Kitty), first recorded in Scottish c. 1400; according to OED it became frequent in English 15c.-16c. Extension to surnames seems to date from c. 1940. Use with common nouns seems to have begun in Scottish with laddie (1546) and become popular in English due to Burns' poems, but the same formation appears to be represented much earlier in baby and puppy.

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.