A “martini shot” is the last shot on a film or the last shot of the day for a film crew—after which the crew can all celebrate with martinis. It’s not known when the slang originated.
 
“‘Come on, this is the martini shot,’ Billy Crystal said” was cited in print in 1985.
   
 
Wikipedia: Martini Shot
Martini Shot is a Hollywood term that describes the final shot set-up of the day. According to Dave Knox, author of the film industry slang guide Strike the Baby and Kill the Blonde, the Martini Shot was so named because “the next shot is out of a glass”, referring to a post-wrap drink.
 
7 December 1985, Marietta (GA) Daily Journal, The Cobb County Times, “Hines, Crystal look marvelous as they film movie in Chicago” by Roger Ebert, pg. 6E, col. 5:
“Come on, this is the martini shot,” Billy Crystal said.
 
Google Books
Ebert’s Little Movie gGossary:
A compendium of movie clichés, stereotypes, obligatory scenes, hackneyed formulas, shopworn conventions, and outdated archetypes
By Roger Ebert
Kansas City, MO: Andrews and McMeel
1994
Pg. ?:
Martini Shot. In a production schedule, the last shot of principal photography at the end of the day is called the “Martini Shot,” after which the crew can retire for the day and, in theory, enjoy a martini.
       
Google Groups: rec.arts.movies.production
Checking for Chickens (was MOS)
Patrick Blackard
5/29/96
(...)
Martini shot meant that you’d soon have one in your hand, in the days when a cooler of “wrap beer” appeared at the end of each day. 
 
Joel Haber
5/30/96
(...)
Well, from all the films I’ve worked on, the term “Martini shot” was used solely to describe the last shot of THE SHOOT.  I guess that on low budget stuff you can’t afford martinis more than once on the shoot!  😉
 
Google Books
The Language of Cinema
By Kevin Jackson
Manchester: Carcanet
1998
Pg. 154:
martini, martini shot recent US industry slang for the final shot of the day; as soon as it’s completed, you can have your martini.
 
Google Books
Uva’s Basic Grip Book
By Michael Uva and Sabrina Uva
Woburn, MA: Focal Press
2002
Pg. 210:
MARTINI SHOT The last film shot of the day. Slang for the next shot will be “in a glass,” or the wrap shot.
 
Google Books
The Pocket Idiot’s Guide to Not So Useless Facts
By Dana Sherwood and Sandy Wood
New York, NY: Alpha Books
2006
Pg. 106:
THe last shot of the day on a movie set is called the “Martini shot” — presumably because the next “shot” that will be had will be part of a cocktail.
 
Google Books
The Grip Book:
The Studio Grip’s Essential Guide (Fifth Edition)
By Michael G. Uva
Burlington, MA: Focal Press
2014
Pg. 531:
MARTINI SHOT Last film shot of the day. Slang for the next shot is “in a glass” (the wrap shot).