It might be a fairly obvious reason, but the name that the president himself used probably was a factor. For example, Kennedy, Johnson, and Truman have their middle initial included on campaign materials as well as in their signature. Arthur, Hayes, and Polk also generally have their middle initial included when their names are used. Bill Clinton and Jimmy Carter rarely used their full first and middle names even in their signature, and the majority of presidents don't use their middle initial.

As for why some are referred to by only initials, I think that's a combination of marketing/branding during campaigns as well as just for brevity. It's similar to Eisenhower using the nickname Ike and Nixon being called Dick on campaign materials. FDR, JFK, and LBJ also roll off the tongue much easier than something like RWR or GRF.