This has a rich history and it is inconclusive where it derives from. There were many great things involving Embassy in the name, such as clubs, cocktail parties, and the cocktail itself. The mention of Embassy cocktail parties appear as early as:
Catalogue of copyright entries: Pamphlets, leaflets, contributions …: Volume 19, Issue 1 – Page 699 by Library of Congress. Copyright Office in 1923
“Embassy club restaurant delicatessen.16528 © Feb. 18, 1922”
LIFE – Feb 26, 1965 – Page 40 Vol. 58, No. 8
“And recently Peking hostesses have been astonished when Chinese officials actually did show up at embassy cocktail parties.”
These type of parties appear in print time and time again, even today. They often were accompanied by Big Bands called Embassy Cocktail Bands. There were many clubs who adopted the name for their establishment, one of the first and most famous is seen here:
Exhibition of pictures by members of the Embassy Theatre Club by Embassy Theatre Club in 1933
Regardless of the elegant parties or famous establishments, this drink does not appear until relatively recently, so perhaps the event that pushed this drink into modern american drinking culture was the Television series as seen here:
The complete directory to prime time network and cable TV shows, … by Tim Brooks, Earle Marsh in 1999
“Club Dance was first telecast in the afternoon on TNN but later moved into the early evening. The show originated from Knoxville, Tennessee. CLUB EMBASSY (Variety) FIRST TELECAST: October 7, 1952 LAST TELECAST: June 23, 1953 BROADCAST.”
It is hard to determine, best hypothesis concludes that it was the TV Show, although it could have easily been named after one of the clubs or lavish parties. This drink currently is known as a “Boston Sidecar”, it is worth mentioning that publications that mention the boston sidecar fail to mention the Embassy Cocktail.
This drink dates between 1949 – 1953 and contained at least as of then:
Shake these ingredients:
Rum (Carico Light as early as 1960)
Strain into a chilled cocktail glass.