It appears that the name was widely used during his administration, and that it was something of a family tradition. Please excuse the extremely nonacademic, but highly specific, source:

As for Franklin, the nickname was stamped on him from birth: A card announcing his arrival in the world on Jan. 30, 1882, depicted a stork carrying a bundle inscribed, “F.D.R.” Young Franklin apparently fancied the appellation, signing “FDR” on a letter he sent to a friend as early as age 9. Newspapers no doubt found the abbreviation especially handy, given that there had already been a President Roosevelt. (“FDR” appeared in the New York Times from the first year of his presidency.) Many of his New Deal programs were similarly stamped with three-letter initialisms (WPA, TVA, SSA, etc.). That may have been a coincidence, or it may have reflected a cultural trend, as companies such as IBM, RCA, NBC, and CBS had risen to prominence in the previous decade. (Three-letter abbreviations of business titles, such as CEO and CFO, came along much later.)