New York’s Palace Theatre at 1564 Broadway (at West 47th Street) was the biggest vaudeville venue. Performers (often waiting for auditions and calls) could be found on the sidewalk in front of the Palace and the small traffic island called Duffy Square across the street—now the location of the TKTS tickets booth. This small piece of real estate was dubbed “Hope Island” by at least 1929.
 
The same “Hope Island” area was also called “Palace Beach” (or “The Beach”).
 
“Palace Beach” and “Hope Island” have also been called “Panic Beach” and “Panic Island.”
 
   
Wikipedia: Palace Theatre (New York City)
Palace Theatre is a Broadway theatre located at 1564 Broadway (at West 47th Street) in midtown Manhattan, New York City. From 1913 through about 1929, the Palace attained legendary status among vaudeville performers as the flagship of the Keith–Albee organization, and the most desired booking in the country.
 
K. Brian Neel’s Vaudeville Terminology
Hope Island - The sidewalk in front of the Palace Theatre.
 
Old Fulton New York Post Cards
23 February 1929, The Vaudeville News and New York Star (New York, NY), “Round the Square” by Arthur Leslie, pg. 7, col. 1:
Where There’s Hope
Hope Island, that triangle on Broadway and Seventh Avenue, between 46th and 47th streets, received its name from wishing artists who inhabit it during good weather and say “Hope I land the last half.”
 
Old Fulton New York Post Cards
18 May 1929, The Vaudeville News and New York Star (New York, NY), “Broadway Baubles” by Paul Dejeneris, pg. 13, col. 6:
Hope Island in Longacre Square is filled with men basking in the sun.
 
23 October 1929, Variety, “Square Described,” pg. 45, col. 4:
A fellow walked up to Alf Grant yesterday and said: “Have you seen Lefty?”
 
Grant replied: “Yeh, I saw him standing in Hope Island on the Waffle iron about ten minutes ago. He walked down Cripple Creek with a couple of turtles and then headed to the water-hole.”
 
“What the devil are you talking about?” inquired the first.
 
Grant then explained that Hope Island was the spot in front of the Palace; the waffle iron was the sidewalk grating; Cripple Creek was 46th, where idle musicians hang out; that turtles were lay-offs standing in the sun and the water-hole was the automat.
 
16 February 1938, St. Louis (MO) Post-Dispatch, “Missouri’s Hillbilly Trio” by William Vaughan, pg. 2D, col. 2:
SPRINGFIELD, Mo., Feb. 16.
ON “Hope Island,” the tiny strip of cement which vaudeville actors have colonized in the center of New York traffic, hoofers, jugglers and comedians turn up their threadbare coat collars against the February breeze which whistles around the Palace Theater and wonder what happened to vaudeville.
   
Google Books
Vaudeville U.S.A.
By John E. DiMeglio
Bowling Green, OH: Bowling Green University- Popular Press
1973
Pg. 122:
The sidewalk outside the Palace was nearly as famous as the theater itself. This stretch of Broadway was dubbed the Palace Beach, usually shortened to Beach. Some also referred to the location as Hope Island. There one could see big names— Jolson, Cantor, Jessel, Rubin, Blue (pick your headliner) — and many small ones. They all spent time on the Beach, swapping stories, showing off their diamonds, or just plain gossiping.