He points out that in Strunk, most of the basic elements necessary to lead to the FANBOYS formulation are there, but the specific list of words is different, and the formulation isn't quite the same. From this, he concludes that FANBOYS postdates Strunk, which is a good guess, but not actually necessary, as Strunk could simply have not known about it. Also (minor point), Brett gives the date of Strunk as 1935, but the original edition was 1918, and the dicta about independent clauses can be found there.
As a regular word, the first citation that the OED has of fanboy in its ordinary meaning is 1919, and then there is a big gap in attestations until 1982. The OED has a call out for interdatings of the word.
Thanks to some recent scans by Google Books, I have found interdatings, and one key one may well get us close to the origins of the acronym:
The earliest source (so far) is Learning to Write by Reed Smith, Bill Paxton, William Paxton, and Basil G. Meserve, 3rd ed. Heath, 1951. Google only gives a snippet view, but that's enough: "[the] chief co-ordinating conjunctions are sometimes called the fanboy words." (p. 398).
The use of the passive voice suggests the acronym was familiar to the authors from their teaching practice. Perhaps the notion had been passed around from teacher to teacher before. Since this book is a third edition, the acronym may go back even further and simply not be shown by Google Books, but the appearance of new co-authors for this edition indicates clearly underwent major alterations, so this could well be the earliest print appearance.
The lack of s implies that so does not form a part of the set for these authors, although other grammar books of the same time or earlier sometimes do call it a coordinating conjunction.
The next work found in Google Books, and the first that actually gives FANBOYS as an acronym, is dated 1970:
Gertrude B. Corcoran, Language arts in the elementary school: a modern linguistic approach, Ronald Press Co. 1970, p. 140:
"Coordinating conjunctions can be remembered by thinking of the acronym FANBOYS"
What's particularly interesting about these appearances is how isolated they are. Few other writers of grammar/composition books seem to have noted the formulation until the 1990s, just at the time when fanboy was gaining wider currency as a term for a passionate fan (e.g., of comic books). Google has one occurrence in a grammatical context from the 1980s (1982). Then there are 8 from the 1990s, and about 70 in the 2000s.
I don't want to make too much of the increasing numbers, because it's hard to be certain that Google has really scanned a representative sample of works from each decade, but the increase does suggest that the acronym became more meaningful to people as the word itself became better known.
The larger question behind the origin of FANBOYS is why these particular words made it into a list. And, but, or, and nor are (if we ignore some niggling exceptions) clear coordinators, but for, yet, and so have significant differences from the pure coordinators.
It would be interesting to trace the development, in schoolbook grammars, of how the set of coordinating conjunctions is defined over time. A cursory reading of several books that precede the fanboys formulation shows that there is a lot of variation, and that there only thing that really seems to recommend fanboys is that it is catchy rather than accurate. That, however, is a story for another post.