Entry in progress—B.P.
     
Wiktionary: on the rocks
Adjective
on the rocks (not comparable)
1.Poured over ice, usually in reference to alcoholic drinks.
Whisky on the rocks
2.In a bad state, particularly of a romantic relationship.
Joe and Tammy’s relationship is on the rocks
 
(Oxford English Dictionary)
rock, n.
slang (orig. U.S.). In pl. Cubed ice for use in an (alcoholic) drink. Freq. in on the rocks (see Phrases 5b).
1946 Amer. Speech 21 35   Rocks, ice.
1959 ‘J. Christopher’ Scent of White Poppies vi. 82   Rocks in your Scotch, Cam? I can get some from the fridge.
 
on the rocks.
orig. U.S. Of an alcoholic drink: served with ice. Cf. sense 6h.
on-the-rocks glass n. = rocks glass n. at Compounds 2a(a).
1949 Life 14 Nov. 63   Ordering a Scotch on the rocks at the bar.
1953 N.Y. Times 14 June 61 (advt.)  On-the-rocks glasses by Hickok.
1955 J. B. Priestley & J. Hawkes Journey down Rainbow 220   They all drank a lot of whisky-on-the-rocks.
   
Google Books
Maigret in Exile
By Georges Simenon, Arthur Maling and Margaret Millar
Roslyn, NY: Published for the Detective Book Club by Walter J. Black
1942?
Pg. 134:
“Your usual Scotch on the rocks?”
 
Google News Archive
1 October 1947, Milwaukee (WI) Sentinel, “Along Amusement Row” by Buck Herzog, pt. 1, pg. 6, col. 2:
Here are some fad drinks: “Scotch on the rocks”—scotch over ice cubes.
 
Google News Archive
31 January 1949, Dubuque (IA) Telegraph-Herald, “Women Imbibe More Than Men,” pg. 6, col. 6:
“They drink anything alcoholic. The stronger the better. They seem to prefer Martinis, boilermakers or Scotch or bourbon ‘on the rocks’ (over ice cubes).”
 
29 August 1952, New York (NY) Times, pg. 4 ad:
Puerto Rican RUM-on-Rocks
1 1/2 oz. Puerto Rican Rum
1 dash Angostura Bitters
Pour this light, dry Rum over ice cubes for the smoothest “On-the-Rocks” you’ve ever tasted.
(Rums of Puerto Rico—ed.)
 
27 October 1964, New York (NY) Times, “Food: On the Rocks at Cocktail Time” by Craig Claiborne, pg. 32:
THE drinking habits of New Yorkers change from restaurant to restaurant, season to season and midday to evening. Their favorite drinks are martinis and Scotch, drunk, more often than not, over ice, or to use the popular phrase “on the rocks,” although less than 10 years ago the thought of drinking cocktails in such a fashion was all but unheard of.
     
One of the three most popular noontime drinks is the bloody Mary, and it is generally served “straight up”—the term that designates the absence of ice.
 
OCLC WorldCat record 
Straight up or on the rocks : the story of the American cocktail
Author: William Grimes
Publisher: New York : North Point Press, 2001.
Edition/Format:  Book : English