Entry from OED Online

Forms: 19- Full Monty, full Monty, full monty. [< FULL a. + monty (origin unknown).
  Many theories are proposed as to the origin of this phrase, but none of them is supported by reliable historical evidence. Perh. the most plausible is that it is from a colloquial shortening of the name of Montague Maurice Burton (1885-1952), men's tailor, and referred originally to the purchase of a complete three-piece suit. Also popular but unsubstantiated is the belief that the phrase is somehow derived from Monty, the nickname of Field Marshal Bernard Law Montgomery (1887-1976). However, the sheer variety of often vague, purely anecdotal, and mutually contradictory explanations for the connection—ranging from his wartime briefing style to his breakfasting habits—renders this less credible. Other suggestions, including references to MONTY n. and MONTE n.¹, are still more speculative.] 

    1. Chiefly with the. Everything which is necessary, appropriate, or possible; ‘the works’.
 
  1985 K. HOWARTH Sounds Gradely (North West Sound Archive), Full-monty, everything included..a thorough display— messing about. 1986 J. MILLER Street Talk 42 Full monty, everything included. To avoid the awkwardness of stumbling through an unfamiliar menu, someone might tell the waiter: ‘We'll have the full monty’. 1994 Guitarist Sept. 169/1 (advt.) When you buy your new Vester bass from us, you can rest assured that it receives the full monty. 1995 Guardian 13 Jan. I. 24/5 When conducting a funeral he wears the full monty; frock coat, top hat and a Victorian cane with metal tip. 2001 Leicester Mercury (Electronic ed.) 13 Jan., What if you're so overworked that when you get back from the office, nothing less than The Full Monty will do when it comes to domestic help?
 

    2. spec. Freq. in form Full Monty. Chiefly with the. Total nudity (esp. for public display); a striptease. Freq. (in later use) in to do the Full Monty: to strip. Also as adj.: totally naked.
  The use was popularized by Peter Cattaneo's 1997 film The Full Monty, about a group of men who become strippers after being made redundant.
 
  1997 S. BEAUFOY Full Monty (film script) 86 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. 1997 Sunday Mirror (Electronic ed.) 12 Jan., Disappointingly, Steve doesn't get his kit off in Ivanhoe. ‘I'm bare-chested a couple of times, but not the full Monty,’ he says. 1999 Evening Express (Aberdeen) (Electronic ed.) 24 May, A hunky TV highlander joined North-east businessmen in a money-spinning Full Monty. 2001 Evening Chron. (Newcastle) (Electronic ed.) 15 Jan. (headline), Bingo boys do the Full Monty—Cheeky callers in sponsored strip for charity calendar.