Jay Gatsby's given and family name at birth was James Gatz, familiarly known as Jimmy. Gatz is a family name of German derivation, which would make it immediately less-than-desirable in the years surrounding and following World War I. As part of reinventing himself into the ideal modern American, he would have easily determined to eliminate that association. Many Jewish family names end with the letter "z," so he may well have had in mind disassociating himself from any possible or perceived connection to the Jewish faith. Choosing "Jay" for a first name eliminates any chance of another juvenile-sounding nickname being created, thus escaping the indignity of being called Jimmy.
In his planning to become a successful American millionaire, he was undoubtedly looking for a name that would be easy to remember and difficult to mispronounce. Jay Gatsby is short, easy to say and spell, and has a nice "ring" to it, all qualities that would have made it desirable.