I believe this is part of the answer. From the NY Times, June 25, 1943:
"The City Council, by a vote of 19 to 6, overrode yesterday Mayor Fiorello H. La Guardia's veto of a local law designating the huge airport under construction in the Idlewild section of Queens as Maj. Gen. Alexander E. Anderson Airport."
And then this, from the NY Times, September 24, 1947:
"Idlewild Airport, renamed 'New York International Airport' by the New York Port Authority, will be ready for international operation next spring, Austin J. Tobin, executive director of the Authority, said yesterday."
Obviously, there are some gaps here. What were the exact dates that each name was in effect? Was it called Anderson Airport up until 1947? Or did it change to Idlewild Airport before then? Or did it have no name at all?
"I tell you this, no eternal reward will forgive us now for wasting the dawn." Jim Morrison