Wordorigins.org

This British phrase, meaning the whole thing, dates to the mid-1980s. It is of unknown origin. From K. Howarth’s 1985 Sounds Gradely:

Full-monty, everything included...a thorough display—no messing about.

The sense meaning total nudity stems from the 1997 film of that title. Prior to this, the full monty did not especially connote nudity. From Simon Beaufoy’s script:

Horse: No one said owt about going the full monty to me...Gaz: We’ve got to give ‘em something your average ten-bob stripper don’t.

While we have no idea where the phrase actually comes from, that hasn’t stopped people from proffering explanations. Here are a few. None of these have any reliable evidence to support them:

• It refers to Field Marshal Montgomery’s habit of meticulously planning his assaults, including intensive and detailed artillery preparations;
• It refers to Montgomery in full-dress uniform with all his medals;
• It refers to Montgomery’s habit of eating a large breakfast each morning;
• Breakfast, but not Montgomery’s, instead it’s the one served by Mrs. Montague at the Lennox Cafe in Bognor Regis, West Sussex;
• It refers to expensive formal clothing purchased at the tailor shop of Montague Burton;
• It is gambler’s slang derived from the game of three-card Monte;
• It is a corruption of the full amount;
• It derives from a television commercial for fruit juice in which an actor asks for the full Del Monte;
• Finally, it could come from Australian and New Zealand slang, a monty being a bet (especially on a horse) that is a sure thing.

(Source: Oxford English Dictionary, New Edition)