“First lady” was lunch counter slang for “spare ribs.” In the biblical story of Adam (the first man) and Eve (the first woman), Eve was created from Adam’s spare rib. “Spare Ribs: One First Lady” was printed in the article “Lunch Room Jargon” in the Middletown (NY) Times Herald on September 9, 1936.
 
Lunch room slang became mostly historical by the 1950s and 1960s.
   
     
Wikipedia: Adam and Eve
Adam and Eve are the Bible’s first man and first woman. Adam’s name appears first in Genesis 1 with a collective sense, as “mankind”; subsequently in Genesis 2-3 it carries the definite article ha, equivalent to English “the”, indicating that this is “the man”. In these chapters God fashions “the man” (ha adam) from earth (adamah), breathes life into his nostrils, and makes him a caretaker over creation. God next creates for the man an ezer kenegdo, a “helper corresponding to him”, from his side or rib. She is called ishsha, “woman”, because, the text says, she is formed from ish, “man”. The man receives her with joy, and the reader is told that from this moment a man will leave his parents to “cling” to a woman, the two becoming one flesh.
       
9 September 1936, Middletown (NY) Times Herald, “Orange Blossoms: Lunch Room Jargon,” pg. 4, col. 4:
Spare Ribs: One First Lady.
 
Google Books
American Speech
1936
Pg. 40: 
Eve with the lid on (apple pie) is an oldster but deserves a place alongside other terms of the Adamic era, Adam’s Ale (water), Adam and Eve on a Raft (poached eggs on toast), and First Lady (spare ribs); ...
   
Google Books
Punch
1939
Pg. 301:
Everyone will profit if you just sit quietly sipping your recently stretched coke or your lately burned malted milk and listen to him refer to such things as
First lady. Spare ribs.
 
28 January 1987, Marshall (TX) News Messenger, “New book spotlights short-order cook terms” by Rhonda Hoeckley (Harte-Hanks News Service, pg. 5B, col. 1:
(The Dictionary of American Food by John Mariani.—ed.)
First lady—spare ribs, a pun on Eve’s being made from Adam’s spare rib.
 
Google Books
A Dictionary of Confusable Phrases:
More Than 10,000 Idioms and Collocations
By Yuri Dolgopolov
Jefferson, NC: McFarland & Company, Inc.
2010
Pg. 121:
first lady, the
(...)
(U.S. diners) spare ribs: The phrase “first lady” meaning “spare ribs” was a pun referring to Eve. The Bible describes her as being made of one of Adam’s ribs.
 
Sarasota (FL) Herald-Tribune
Diner language
By Linda Brandt
Posted Jan 12, 2011 at 12:01 AM
(...)
Besides the earlier raft reference, Adam and Eve inspired “Eve with a lid,” for apple pie (presumably because Eve tempted Adam with an apple). The lid is the top crust. Add cheese, and you have “Eve with a moldy lid.” “Adam’s ale” is slang for water. And because the Old Testament says God fashioned Eve from Adam’s spare rib, an order for spare ribs translates to “first lady.”
 
Google Books
Hash House Lingo:
The Slang of Soda Jerks, Short-Order Cooks, Bartenders, Waitresses, Carhops, and Other Denizens of Yesterday’s Roadside
By Jack Smiley
Introduction by Paul Dickson
Mineola, NY: Dover Publications, Inc.
2012 (Originally published in 1941)
Pg. 62:
First lady—spare ribs
An allusion to the extra rib that Adam gave woman in Genesis. A 1912 compilation of restaurant slang contained this call: SHORTY BROWN IN SWIMMING for short ribs in brown gravy.