The “Big Street” is a nickname for the Broadway theater district in New York City, also called the “Great White Way.” Sheppard “Shep” Friedman (1875-1921), an editor of the New York (NY) Evening Telegram and New York Morning Telegraph, was credited in 1926 for coining both terms.
   
“J. W. Snyder of Erie, Pa., who is staying temporarily at No. 58 Greenwich street, this city, is much sadder but slightly wiser since his experience with a quickly made friend in the ‘big street,’ as he calls Broadway” was printed in the Elmira (NY) Daily Gazette and Free Press on September 29, 1900. “Foolish Happenings to Real Folks Along the Big Street” was printed in The Sunday Telegraph (New York, NY) on August 21, 1904. “The Big Street never looked just the same without him” was printed in The Morning Telegraph (New York, NY) on January 10, 1905.
 
“Big Rue” is a similar Broadway nickname that was used in the 1920s.
         
The Big Street (1942) is a movie starring Henry Fonda and Lucille Ball, based on the short story “Little Pinks” by Damon Runyon. Interstate highways were constructed in the 1950s and the “Big Street” nickname has been only infrequently used in the years following the movie.
     
       
Wikipedia: The Big Street
The Big Street is a 1942 American drama film starring Henry Fonda and Lucille Ball, based on the short story “Little Pinks” by Damon Runyon, who also produced the movie. The film was directed by Irving Reis. The screenplay was written by Leonard Spigelgass from Runyon’s story.
   
Name’s source
The Big Street was a nickname for Broadway, where this movie’s plot starts, and where all Runyon’s stories take place.
       
29 September 1900, Elmira (NY) Daily Gazette and Free Press, “How Snyder Lost His Roll,” pg. 2, col. 5:
New York, Sept. 28.—J. W. Snyder of Erie, Pa., who is staying temporarily at No. 58 Greenwich street, this city, is much sadder but slightly wiser since his experience with a quickly made friend in the “big street,” as he calls Broadway.
 
Old Fulton NY Post Cards
1 June 1902, The Sunday Telegraph (New York, NY), “Gossiping Boulevardiers Cut Wide Swath Along Broadway’s Path of Summer Dalliance” by George L. Macfarlane, pt. 2,  pg. 2, col. 4:   
Therefore, the big street is filled with men who wander about aimlessly in an effort to avoid housecleaning at home, and with women who have taken an hour or two off from the grind of the annual Spring clean-up to run over and see “what is doing” on Broadway.
 
Old Fulton NY Post Cards
21 August 1904, The Sunday Telegraph (New York, NY), pt. 2, pg. 5, col. 1:
DOWN THE LINE
Foolish Happenings to Real Folks Along the Big Street, Between the Flatirons. Recorded by the Solemn Guy.
 
Old Fulton NY Post Cards
10 January 1905, The Morning Telegraph (New York, NY), “Actor Revelle Is Resurrected,” pg. 1, col. 2:
Broadway has never been quite easy in its mind since this gifted artist disappeared, after his engagement with Mrs. Leslie Carter in “Du Barry.” The Big Street never looked just the same without him.
 
Google Books
10 October 1908, New York (NY) Star, pg. 19, col. 2:
What the New York Critics Say
TELEGRAPH: “Nine of the handsomest show girls that ever appeared on the big street.”
 
Newspapers.com
8 November 1908, Brooklyn (NY) Daily Eagle, sec. 2, pg. 5, col. 5 ad:
PRESENTING LOCALS ALONG THE BIG STREET
NIGHT ON BROADWAY
 
Google Books 
The Girl that Goes Wrong
By Reginald Wright Kauffman
New York, NY: Macauley Company
1911
Pg. 181:
Pen had long since ceased to be Broadway’s idol; she now became its joke. The Big Street’s population changes yearly, and the newcomers knew not Penelope.
 
Google Books
The Least Resistance
By Kate L. McLaurin    
New York, NY: George H. Doran Company
1916  
Pg. 32:
Then the talk grew more personal, it was of their past achievements, present grievances and future hopes. All that any of them asked was a fair chance on Broadway.
 
“If the Big Street says I’m no good, I’ll quit,” Harmon said, shifting his pipe.
 
Google Books
September 1919, The Wide World Magazine, “The Adventures of a Newspaper-Man” by Frederic Martin, pg. 435, col. 2:
It was in the Mills Hotel that I met the sturdy vetera of the Civil War whose principal aim in life seemed to be to convince me that Broadway is the only really “big street” in the world—that it is the longest, busiest, and handsomest thoroughfare the world has ever seen.
 
25 March 1921, New York (NY) Tribune, pg. 13, col. 7:
Sheppard Friedman, N.Y. Newspaper Man, Is Dead
Served Several Metropolitan Dailies and Was Theatrical Advance Man
Sheppard Friedman, newspaper and theatrical man, died yesterday at Mount Sinai Hospital from pneumonia. He was born in Austin, Tex., forty-seven years ago. His parents moved to Fort Worth, where he began his newspaper career by selling papers.
 
He became a reporter on the staff of The Fort Worth Gazette. In 1900 he came to New York. He had been on the staffs of The Evening Telegram, The New York Herald, The Morning Telegraph, the Hearst newspapers, The Evening World and The World. His last position was on the staff of The World. He was an able newspaper man and has served capably also as advance agent for A. H. Woods and Arthur Hammerstein. His mother and eight brothers survived him. His wife died five years ago.
     
25 March 1921, New York (NY) Times, pg. 15:
Sheppard Friedman.
Sheppard Friedman, long a member of the staff of The World, as a reporter, copy editor, and latterly as a rewriter of local stories, died yesterday in Mt. Sinai Hospital of pneumonia. He was 47 years old, and after reporting for the Fort Worth Gazette, he came to New York in 1900. Mr. Friedman had been with The Herald, Telegraph, Telegram and Evening World, but most of his journalistic career here was with The World. He was also well known in the theatrical profession, having acted as publicity agent for various productions.
 
Newspapers.com
8 July 1921, The Black Dispatch (Oklahoma City, OK), pg. 1, col. 5:
COLORED QUARTETTE SETS THE MUSICAL WORLD A-TALKING
(...)
The latest “find” if the Norfolk Jazz Quartette, a colored aggregation who have come from Norfolk, Va., to the “Big Street Broadway.”
 
5 December 1926, New York (NY) Morning Telegraph, pg. 20, col. 1:
Do You Remember?
The origin of that phrase, “the great white way,”—one of the most allusive and illusive phrases in the American tongue—was revealed on this page last week, thanks to that enterprising journal, The New York Press. The Press, as quoted here, said in part:
 
That name was tacked on Broadway by a Morning Telegraph man more than twenty-five years ago. Shep Friedman, cleverest, brightest and most sophisticated of the newspaper men of his day, so named the street, and the name clung because it belonged.
 
Now comes this interesting letter from that beau ideal of press representatives, Arthur J. Levy, of the House of Belasco, who writes:
 
“Dear Charley Parmer:”

It’s mighty good to know that some one is left in the newspaper game that remembers Shep Friedman. Good old Shep, what a jolly time he must be having now looking down upon some of the gang he knew and laughing that dry kindly laugh of his. Do you remember that laugh? I can hear it now. No one could laugh like Shep, with you or at you. Many times he has razzed a story of mine. But the knowledge of “The Big Street” he crammed into me is responsible for any little success I have achieved in the theatre.
 
“I heard a well-known film P. A., one of whom we both know, tell how he first termed Broadway “The Great White Way.” But the few oldtimers left, and thank heavens, a few are still left, will always remember it was Shep who first gave the greatest thoroughfare in the world the name that it is known by in every nook and corner of the universe.
 
“It was he too, that called it “The Big Street.” If you would start out to find a list of the bright sayings coined by Shep you would only have to seek the book of most any musical comedy of the day—the successful ones I mean—and you would find a heap of librettists, who have also remembered Shep—only they don’t want any one to think so.
 
“So, if in your peregrination up and down the street that Shep loved so much, you find a saying here and there that you know Shep coined, let us have it. You will find a few of us have not forgotten Shep and we still treasure his quaint old ways. But where ever Shep is, he is happy. He would not know how to be otherwise.
 
“Thanks for the good thought. He sure was a regular guy.
“Earnestly,
(Signed) “Arthur J. Levy.”
 
Those of you who knew Shep Friedman well, and remember the glittering original phrases which he tossed off while wandering up and down the street, won’t you write them out and send them in?
 
Newspapers.com
2 January 1927, Fort Worth (TX) Star-Telegram, pg. 2, cols. 1-4:
FACILE PEN OF FORT WORTH
REPORTER GAVE BROADWAY
NAME OF “GREAT WHITE WAY”
BY W. G. BYRNE. (Originally published in the New York Morning Telegraph,  on November 28 and December 5, 1926.—ed.)
New York’s one and only Broadway, the street that has been praised or damned (as the occasion warrants) in endless reams of prose and poetry, received its best known appellation “Great White Way” from the ready pen of a former Fort Worth newspaper man—Sheppard S. Friedman.
 
Although Friedman coined the phrase more than 25 years ago and Friedman himself for more than six years has been buried in Fort Worth Hebrew Rest here with others of the Friedman family who have passed on, New York only recently awakened to the fact that someone was responsible for the name.
 
Most Broadway-ites had apparently taken a Topsy-like attitude toward the matter—it had simply “growed.” Or possibly they never gave it a thought. But at any rate although “Shep” has long since been gathered to his fathers, he is now duly accredited by New Yorkers for having tacked on to the street a name more euphonious than even Broadway or “The Big Street,” a name for Broadway also accredited to Friedman.
 
14 August 1927, Sunday News (New York, NY), “About Broadway” by Mark Hellinger, pg. 25, col. 5:
That’s the way they go along the big street.
 
Playbill
Connie Grappo Attached to Menken Musical, Big Street
OCT 08, 2002
Big Street, a new musical with music by Alan Menken and lyrics by Marion Adler, now has a new book by Connie Grappo, who is also attached to the project as director.
(...)
The show is based on the 1942 Lucille Ball-Henry Fonda film, “The Big Street,” which was, in turn, drawn from the Damon Runyon story “The Little Pinks.” Busboy Little Pinks loves nightclub singer Gloria from afar, but gold digging Gloria pays him no mind, setting her sights only on the most moneyed sugar daddy. Irving Reis directed the original film, which featured Ball as Gloria, Fonda as Little Pinks, and character actor Eugene Pallette as Nicely Nicely Johnson, another Runyon figure familiar from the musical, Guys and Dolls.