According to English-for-Students.com, New Orleans' nickname "The Big Easy" refers to its laid-back attitude and the easy-going nature of the jazz musicians and other residents of the city. The name came into common use after the release of James Conaways' novel entitled "The Big Easy" in 1970, but its exact origins remain unclear.
English-For-Students.com explains that Conaways likely picked up the nickname "The Big Easy" from slang what was already being used in the New Orleans area. Some also report the existence of a jazz club called "The Big Easy" in the early 1900s, and it's possible that the nickname evolved from that of the club. However, there is no evidence to prove that this club existed.
There are said to be ties between the nickname "The Big Easy" and New York City's nickname, "The Big Apple," with some claiming that New Orleans' nickname was coined in order to contrast the city with the more energetic New York. Nutrias.org states that in the 1970s, a local gossip columnist wrote an article contrasting New York and New Orleans, and used the term "The Big Easy" as a metaphor for the laid back lifestyle to be had in the latter city. The nickname became more commonplace in the years that followed.