“Aspirin Alley” is an infrequently used nickname for Broadway, possibly because the street’s dizzying lights require a person to take an aspirin. New York newspaper columnist O. O. McIntyre (1884-1938) used “Aspirin Alley” in a column on November 15,  1929.
 
The blog Gothamist reported in 2012 that One Mile House (10 Delancey Sreet, between Bowery and Chrystie Street) served a cocktail called “Aspirin Alley,” made from Tanqueray, Creme De Violette, Maraschino Liqueur and Cremant De Loire.
 
   
Wikipedia: Broadway (Manhattan)
Broadway /ˈbrɔːdweɪ/ is a road in the U.S. state of New York. Perhaps best known for the portion that runs through the borough of Manhattan in New York City, it actually runs 13 mi (21 km) through Manhattan and 2 mi (3.2 km) through the Bronx, exiting north from the city to run an additional 18 mi (29 km) through the municipalities of Yonkers, Hastings-On-Hudson, Dobbs Ferry, Irvington, and Tarrytown, and terminating north of Sleepy Hollow in Westchester County.
 
Google News Archive
15 November 1929, Miami (FL) Daily News and Metropolis, “New York Day by Day” By O. O. McIntyre, pg. 6, col. 4:
New York, Nov. 15.—Thoughts while strolling: (...) The blinking sign of Zit’s theatrical weekly. The incessant murmur of split weeks, lucky breaks, cluck tours and cheap gin. Aspirin Alley!
 
Old Fulton NY Post Cards
10 November 1943, Amsterdam (NY) Evening Recorder, “Lights of New York” by L. L. Stevenson, pg. 4, col. 5:
A big Marine telling his fair young companion that Broadway should be called “Aspirin Alley”...and when she innocently asks him why, replying, “If you’d been with me last night you’d know.”
 
Google News Archive
24 July 1963, Reading (PA) Eagle, “Answers to Questions,” pg. 10, col. 2:
Q. What are some nicknames for Broadway in New York City? I.T.
A. Some commonly used nicknames are: The Alley, Aspirin Alley, Big Artery, Coffee Pot Canyon, Gay White Way, Dirty White Way, Hardened Artery, The Lane, Main Artery, Main Drag, Main Stem, Mazda Lane, Neon Boulevard, The Big Street.
   
Gothamist
Brews On The Bowery Courtesy of One Mile House
The Lower East Side is getting a taste of the ‘20s thanks to newcomer One Mile House, which recently opened on Delancey Street. The name comes from an 1800s watering hole one mile from City Hall— city politicians who wanted to booze while they “governed” could walk there and imbibe on the clock. This new incarnation keeps the spirit of the past alive with antique subway tiles, tin ceilings and bathroom walls adorned with newspaper clippings from the ‘20s and ‘30s. The spacious bar area has large windows that bring in plenty of sunlight during the day; candles and orb-shaped light fixtures create an intimate atmosphere at night.
(...)
For the stronger stuff, the delightfully named cocktails—Giggle Juice or Aspirin Alley, anyone?—are sure to please your inner drunk, in particular their “Baby’s Got Temper,” a Don Julio tequila-infused take on a classic michelada.
10 Delancey Street // 646-559-0702
Nell Casey in Food on Mar 25, 2012 3:10 pm
   
Foursquare
One Mile House
10 Delancey St (btwn Bowery & Chrystie St.), New York, NY 10002
(...)
Cocktail: Aspirin Alley
Tanqueray
Creme De Violette
Maraschino Liqueur
Cremant De Loire