Sunny Side Up

“Sunny side up” means an order of eggs fried on one side only. The term “sunny side up” has been used in restaurant slang since at least 1887.
 
According to an 1899 newspaper article on restaurant slang (see below), “sunny side up” might have been coined at Michael Casey’s restaurant on New York’s Bowery.
 
 
Wiktionary: sunny side up
Adverb
sunny side up ‎(not comparable)
1. (US) Of an egg, fried on one side, served with the unbroken relatively soft yolk on the top.
   
(Oxford English Dictionary)
sunny side up: of an egg, fried on one side only; hence sunnyside egg.
1901   Dial. Notes 2 149   Sunny side up,..of eggs, to fry [sic] on only one side.
 
Chronicling America
5 March 1887, Lancaster (PA) Daily Intelligencer, pg. 3, col.
Specimens of Slang.
From the St. Louis Globe-Democrat.
Men who write books about slangmight find material in some restaurants. In Kansas City there is an abundance of it. Only in one place in St. Louis—on Morgan street—can you hear downright slang in the giving of orders, and curious it is. If you tell the waiter you want an oyster stew, he shouts out to the kitchen, “Jesse James!” A beefsteak becomes “slaughter in the pan;” plain, black coffee is “coffee in the dark;” potatoes unpeeled are “Murphy with his coat on;” two fried eggs on one side are transformed into “sunny side up;” buckwheat cakes are spoken of in gambler fashion as “stack of reds with copper on top,” and butter cakes as “stack of whites.”
 
Chronicling America
8 July 1887, San Saba (TX) News, “Pith and Point,” pg. 2, col. 8:
Young man (in Park Row coffee and cake saloon)—“Waiter, I want a beefsteak, unpeeled potatoes, and a couple of eggs fried on one side only.” Waiter (vociferously)—“Slaughter in the pan,” “a Murphy with his coat on,” an’ “two white wings with the sunny side up.”—Puck.
 
3 March 1889, The Leader and Herald (Cleveland, OH), “The Theaters,” pg. 11, col. 4:
Herbert Kelsey, the aesthetic leading man of the New York Lyceum Company, rarely goes tp a resort or a restaurant. The other evening he was out late, and, feeling hungry, stepped into a Sixth avenue eating house and called for a steak, baked potates, and two eggs, fried only on one side. The waiter, knowing his customer, sang out: “Slaughter in the pan, two Murphies wid der coats on, and two white wings with the sunny side up.” Mr. Kelsey lost his appetite and was taken home in a hansom.—Philadelphia Telegraph.
 
3 November 1891, The Evening Repository (Canton, OH), pg. 5, col. 3:
RESTAURANT WAITERS’ SLANG.
(...)
White wings on a shipwreck means two eggs turned over or the yolks broken.
 
White wings with sunny side up, two! means two fried eggs with the yolks up.
   
Chronicling America
28 March 1896, The Sun (New York, NY), “Southern Lunch Counter Slang,” pg. 2, col. 6:
Fried eggs unturned are called “eggs with eyes open,” “sunny side up,” “straight up,” and “two white wings turned down.”
 
17 July 1899, Atlanta (GA) Constitution, pg. 5, col. 3
STORY OF A QUEER
CAFE IN NEW YORK
(...)
Michael Casey, the original owner of the place which he styled a cafe, belonged to that class of men known in New York years ago as the “Bowery Boys.” Casey was a prominent member of this peculiar clan and up to the day of his death, which occurred a number of years ago, he always mentioned his connection with the boys as a matter of pride.(...)(Col. 4—ed.)
 
If one sat down to the table and ordered chops and eggs the order went to the cook as: “A stack of reds and two in the air,” and while lost in wonderment and vainly endeavoring to find out what he meant, down would come the dishes with a meal equal to anything at the big hotels.
 
“A dozen in the grease” meant fried oysters; “one jamoca” was for a cup of coffee; “pompano for fifty,” which would undoubtedly cause you to clutch your purse and run, meant simply a half-dollar order of fish; “pork and—,“translated was, “bring beans on the side,” whole “ham and—straight up” gave the patron ham with eggs that were soft on top.
 
“Shipwreck two” was the alarming order for scrambled eggs and “hand me down the B. and O.” was for steak smothered in onions. If you fancies two softboiled eggs the waiter would call out: “Drop two in the well and let ‘em come up easy.” “Plate mystery” brought plain corned beef hash, and if one only desired sausage, “three links of the cable line” brought the dish in a hurry.
 
For mince pie with sugar sprinkled on the top, the order was, “Dyspepsia in a snowstorm.” This term was sometimes changed to “Put raisins in the hash.” An order for eggs on toast went to the kitchen as, “Adam and Eve on a raft,” but if after giving this order the customer wanted the eggs plain, the countermand went out as, “Save Adam and Eve; sink the raft.” Eggs fried on one side were alluded to as, “White wings; sunny side up.”
 
Casey himself was the originator of these unique and peculiar orders and when asked for an explanation as to why he used such an outlandish system, he always wagged his round head, closed his left eye and emitted the one word: “Advertisement.”
 
It was a big advertisement for the house and probably brought many a collar to the Casey coffers that otherwise might have strayed afield.
 
There will be many to mourn the blotting out of this celebrated place, but there is a new Bohemia in Gotham and the fickle public is always prone to forget familiar places at short notice.
C. J. K.
 
27 December 1908, Kalamazoo (MI) Gazette, “Scrambled English Is Served in the Public Eating Places of Kalamazoo,” pg. 10:
You may order eggs fried on one side and hear “sunny side up.”