The nickname was created before they played a single official game. During that first expansion year, the Mets were amazingly bad.
There is little doubt that this was said by Mets manager Casey Stengel. It was repeated and popularized by New York Daily News sportswriter Dick Young. This is the earliest I found.
29 March 1962, New York Daily News, pg. 78, col. 1:
Kanehl's Double Keys Mets
To 4-3 Win in 9th Over LA
By Dick Young
St. Petersburg, Fla., March 28 -- They're a strange conglomeration, these amazin' Mets.
(...)
(Col. 2 -- ed.)
THE 3,192 MEMBERS of the Stengel fan club went wild. They are enjoying these late-inning heroics of the Amazin' Mets, almost expecting them.
(Above the col. 2 box score is "Slightly Amazin'"- - ed.)

