The Longacre Square area of Manhattan—called Times Square since 1904—didn’t have theatres in the 1880s and early 1890s. The crime-ridden area was known as the “Thieves’ Lair.”
 
The Olympia Theatre (1514-16 Broadway at 44th Street) was opened in 1895 by Oscar Hammerstein I, and the famous Broadway theatre district made the old name largely forgotten.
 
     
Wikipedia: Times Square
Times Square is a major commercial intersection and neighborhood in Midtown Manhattan, New York City, at the junction of Broadway and Seventh Avenue, and stretching from West 42nd to West 47th Streets. Brightly adorned with billboards and advertisements,
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As more profitable commerce and industrialization of lower Manhattan pushed homes, theaters, and prostitution northward from the Tenderloin District, Long Acre Square became nicknamed the Thieves Lair for its rollicking reputation as a low entertainment district. The first theater on the square, the Olympia, was built by cigar manufacturer and impresario Oscar Hammerstein I. “By the early 1890s this once sparsely settled stretch of Broadway was ablaze with electric light and thronged by crowds of middle- and upper-class theatre, restaurant and cafe patrons.”
 
Google Books
My Sixty Years in Show Business;
A Chronicle of the American Theater, 1874-1934
As told by George Blumenthal to Arthur H. Menkin
New York, NY: F.C. Osberg
1936
Pg. 137:
Just as his Olympia had changed “Thieves’ Lair” to “Times Square,” so that district (in London—ed.) became known as “Kingsway.”
   
New York (NY) Times
September 05, 1937
NEW SIGHT ON A FAMOUS OLD SITE
[ DISPLAYING ABSTRACT ]
THE opening Thursday of the new International Casino on the east side of Broadway, between Forty-fourth and Forty-fifth Streets, marks another chapter in the history of a block which has played an integral part in the theatre annals of New York.
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... in this neighborhood, the section of Broadway which lies between Forty-second and Fifty-ninth Streets was known as “Thieves’ Lair,” and respectable burghers did not venture there after the sun went down. But the Olyrnpia marked ...
 
Google News Archive
29 March 1950, Tuscaloose (AL) News, “Pitching Horseshoes: New York’s Hissing Case” by Billy Rose, pg. 4, col. 3:
He (Oscar Hammerstein I—ed.) made up his mind to build a theatre which would put Koster and BIal out of business, and unable to find a suitable site on Herald Sqaure he bought a plot on 42nd Street, an uptown morass known in those days as “The Thieves’ Lair.” “Location doesn’t matter,” he told his friends. “It’s what you put in a theatre that counts.”
 
Well, it turned out he was as right as dollar whiskey. Within a fre years, the new theatre, the Olympia, was the most talked-about playhouse inthe country, and before long other theatres were springing up around it to cash in on its turnaway crowds.
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Forty-Second Street these days is again a Thieves’ Lair, studded with hot-dog emporia and flea circuses, and it would be a nice twist if Oscar II were to book his next musical, “Anna and the King of Siam,” in the New AMsterdam theatre and tone up the old neighborhood in deference to the memory of his fabulous grandpappy.
 
Google Books
The Routledge Guide to Broadway
By Ken Bloom
New York, NY: Routledge
2007
Pg. IX:
In 1895, in the area known as “thieves lair” by locals, Oscar Hammerstein I built his Olympia theater complex.
 
New York (NY) Times 
New York Kids’ Stories
Selina Alko’s ‘B Is for Brooklyn,’ and More
By PAMELA PAUL AUG. 8, 2012
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As the grid moved northward up the island, McKendry tells us, the intersection between Broadway and Seventh Avenue became known as Long Acre Square, commonly referred to as “the Thieves’ Lair.” (Some things don’t change.) Of course, its new name, which came along with the city’s first subway line, the completion of the Astor Hotel and – ahem – the new headquarters for The New York Times, was a lot more enduring.