A “hangover” is a result of drinking too much; a “hangover breakfast” is the first meal taken after a night’s drinking spree. The term “hangover breakfast” is cited in print by at least 1933; the famous “21” Club was cited by Broadway columnist O. O. McIntyre for offering a “hang-over breakfast” in 1937.  The “hangover breakfast” is not necessarily any different than any other breakfast. Many restaurants offer a New Year’s Day “hangover breakfast.”
 
“Hangover brunch”—the same concept as the “hangover breakfast, but scheduled several hours later in the day—is cited from the 1950s.
 
 
24 January 1933, Greeley (CO) Daily Tribune, pg. 4, col. 6:
Miss Ida Marie Lorang, daughter of Mr. and Mrs. A. P. Lorang, spent the week in Boulder, being a guest Friday night at Sig Ep dinner dance at Blanchard lodge andthe Junior Prom following. Saturday at 11 o’clock she attended the “hangover breakfast” at the Kappa Alpha Theta sorority house and in the evening was among the dancers at the canon, completing the week end of Junior Prom festivities.
 
3 November 1937, Paris (TX) News, “On Broadway” by O. O. McIntyre, pg. 4, col. 2:
The widely exploited “No. 21,” incidentally has decided to remain open Sunday chiefly to serve what it calls a “hang-over breakfast”—a menu calculated to startle the jaded appetite into accepting a few tid-bits. The most famous of the pick-me-up breakfasts used to be served at Murray’s on West 42nd street and was said to be the creation of the actor, John Mason. It consisted first of chilled orange juice with a coupcon of brandy, then two raw eggs doctored up with Worcestershire sauce and dashes of paprika, then—of all things—marinated herring. It was a favorite of Nat Goodwin, Wilton Lackaye and DeWolf Hopper when the going had been rough.
   
21 September 1941, New York (NY) Times, “The Current Scene” by Robert Van Gelder, pg. BR27:
SO IT DOESN’T WHISTLE. By Robert Paul Smith. 234 pp. New York; Harcourt, Brace & Co. $2.50.
IN many ways this is a good novel of the New York of the rootless young through these last years. Mr. Smith writes of what he knows—of jaz bands, drunks, marijuana smokers, a restaurant and a music shop in the Spanish section, a small East Side bar, a girl who thoughtfully pronounces forthright phrasing to be cute, crap games in back rooms, hangover breakfasts, the poetry of Ernest Walsh, how to kill time around a radio studio, how to put a new band over with the kids.
 
26 December 1944, Chicago (IL) Daily Tribune, “Today with Women” by Judith Cass, pg. 13:
Masculine members of the Key club who attend a “hangover breakfast” on New Year’s day will wear their gayest new Christmas…
 
1 January 1945, Chicago (IL) Daily Tribune, “Chicago Drinks Toast Today to Speedy Victory” by Judith Cass, pg. 25:
One of the gayest of the New Year’s day parties is a “hangover breakfast” to be held at 1 p. m. at the Key club.
 
Time magazine
Monday, Mar. 07, 1949
Package Deal. In Nashville, the Hide-Away Restaurant advertised: “Hangover breakfast . . . tomato juice, raw eggs, aspirin, black coffee, our deepest sympathy.”
 
28 June 1953, Hartford (CT) Courant, pg. A4:
Hangover Breakfast Offered
   
Google Books
The American Thesaurus of Slang:
a complete reference book of colloquial speech
By Lester V. Berrey and Melvin Van den Bark
Edition: 2
New York, NY: Thomas Y. Crowell
1953
Pg. 100 (also Pg. 117):
hangover breakfast, a late breakfast the day after a spree.
 
Google Books
A Room in Paris
By Peggy Mann
Garden City, NY: Doubleday and Company, Inc.
1955
Pg. 275:
The menu listed Hangover Breakfast: tomato juice, two raw eggs, black coffee, aspirin, and our deepest sympathy.