A sedative (or sometimes in the US a purgative) drug surreptitiously slipped into someone's drink.

A 'Mickey Finn', which is sometimes called just a 'Mickey' is supposed to be named after a character from 19th century Chicago - 'Mickey Finn', of course. Finn was the keeper of Chicago's Lone Star Saloon in the late 19th and early 20th century. He was alleged to have drugged and robbed his customers. There are a couple of US newspaper references from December 1903 that allude to this:

Chicago Daily News - "The complete defense advanced by 'Mickey' Finn, proprietor of the Lone Star saloon ... described ... as the scene of blood-curdling crimes through the agency of drugged liquor."

Inter-Ocean (Chicago) - "Lone Star Saloon loses its license. 'Mickey' Finn's alleged 'knock-out drops' ... put him out of business."

Mickey Finn would have been a common enough name in Ireland and amongst Irish emigres to the USA. Ernest Jarrold was an author in late 19th century USA who wrote a popular series of newspaper stories called the Mickey Finn stories, from the early 1880s onward. The main character was a small boy and the stories are in the same vein as Twain's Huckleberry Finn (pub. 1884). It has been suggested that Twain, who knew Jarrold, plagiarized the idea from the 'Mickey Finn' series. Jarrold later wrote under the pseudonym 'Mickey Finn' and the name became a generic term for any Irishman - much like 'Paddy' today.

So, by 1903 there could well have been many people called or known as 'Mickey Finn'. Although Jarrold's, a.k.a. Mickey Finn's, story is interesting and predates the Chicago Mickey Finn's activities, there isn't anything to explicitly link him to the phrase. The only version of the story with any real supporting evidence is that of the Chicago saloon-keeper.