The National Security Agency was formed in 1962 to improve foreign intelligence within the United States government. The agency was so secret (especially in its early years) that even its existence was denied.
 
The NSA nickname/motto “Never Say Anything” has been cited in print since at least 1961. The NSA nickname “No Such Agency” has been cited in print since at least 1982.
 
The NSA nicknames of “Now Spying on Americans” and “National Socialist Agency” have been cited in print since at least 2005 and 2006, but were popularized in 2013 after the NSA’s surveillance programs made international headlines. Other post-2013-spying-scandal nicknames include “National/Neocon/Nefarious Spying Agency,” “National Snoops’ Agency” and “Not Serving Americans/Anyone.”
     
   
Wikipedia: National Security Agency
The National Security Agency/Central Security Service (NSA/CSS) is a cryptologic intelligence agency of the United States Department of Defense responsible for the collection and analysis of foreign communications and foreign signals intelligence, as well as protecting U.S. government communications and information systems, which involves cryptanalysis and cryptography.
 
The NSA is directed by at least a lieutenant general or vice admiral. NSA is a key component of the U.S. Intelligence Community, which is headed by the Director of National Intelligence. The Central Security Service is a co-located agency created to coordinate intelligence activities and co-operation between NSA and other U.S. military cryptanalysis agencies. The Director of the NSA serves as the Commander of the United States Cyber Command and Chief of the Central Security Service.
 
NSA’s work is limited to communications intelligence; it does not perform field or human intelligence activities. By law, NSA’s intelligence gathering is limited to foreign communications, although incidents such as the NSA warrantless surveillance controversy have occurred.
     
8 January 1961, Baltimore (MD) Sun, “American’s Supersecret Watchdog: NSA At Fort Meade Has ‘Unstated National Security Mission,’ Especially In Communications” by John C. Schmidt, pg. A1:
Frustrated newspaper men have referred to the to the NSA initials as meaning “Never Say Anything.”
     
Google Books
Cloak and Cipher
By Dan Tyler Moore and Martha Stifler Waller
Indianapolis, IN: Bobbs-Merrill
1962
Pg. 89:
Insiders will tell you that the initials “NSA” stand for “Never say anything.”
 
4 February 1968, Boston (MA) Globe, “The Pueblo and automated spying” by John Davy and Andrew Wilson, pg. A1:
It is much more hush-hush than the CIA, and the initials NSA are sometimes said to stand for “Never Say Anything.”
 
Google Books
Secret Armies:
The growth of corporate and industrial espionage
By Jacques Bergier
Indianapolis, IN: Bobbs-Merrill
1975
Pg. 169:
Washington gossip has it that NSA really stands for Never Say Anything.
 
20 April 1980, Los Angeles (CA) Times, “Keeping Secrets in the Computer Age” by David Kahn, pg. E5: 
Its initials are said to stand for “Never Say Anything.”
     
Google Books
The Puzzle Palace:
A report on America’s most secret agency
By James Bamford
New York, NY: Penguin Books
1983, ©1982
Pg. 357:
Some Washington wags have been known to say the initials stand for No Such Agency; those inside the wall have another definition: Never Say Anything.
 
New York (NY) Times
WASHINGTON TALK: N.S.A.; Slamming The Press, In Daylight
By STEPHEN ENGELBERG
Published: October 14, 1987
For the National Security Agency, the Government department that intercepts overseas communications, breaks codes and barely acknowledges its own existence, secrecy is more than a way of life. It is an obsession.
(...)
A recent N.S.A. security bulletin illustrates the depth of the agency’s distress over recent publications. The bulletin likens the damage done by several authors who have written about the agency to the harm caused by Soviet spies. And it warns agency employees not to disclose where they work, saying in part: ‘‘You have no doubt heard that N.S.A. stands for ‘No Such Agency’ or ‘Never Say Anything,’ references to the extremely low profile this agency once enjoyed. Unfortunately, because of people like James Bamford, Seymour Hersh and Ronald E. Pelton, N.S.A. has received a great deal of unwanted media exposure in recent years.’‘