I don't know about how it all started. But looking back on some discussions of grading from about 100 years ago, we might get some clues about why a 5-point system has been so common. This comes from the works of several scholars who were writing about grading around 1908-1913: Dearborn at Wisconsin, Finkelstein at Cornell, and Meyer at Missouri.

They wrote about how letter grades were supposed to be distributed, and explored on their campuses what was happening in practice. One of the justifications for a five-point scheme is the idea that students (or, students' performance, or students' attainment) should be distributed along a relatively normal curve, and that a five-division system is sufficient to differentiate students in such a fashion. So, most kids get C's (which are, in this sense, average).

The other justification for having a small number of categories (like 5) helps ensure reliability. That is, if you had a 100-pt grading scale, and asked a number of evaluators to grade a paper or exam, they might deviate in their grading--you might give it a 75, I might give it an 79, the notoriously tough moderators here give it a 71. But if we had to assign a letter grade, we'd all land on C. And reliability in grading is a desirable thing within and between schools.

I don't know why A, B, C became standard, but it wouldn't surprise me to hear that it was no more complicated than just starting with the first letter of the alphabet and going through the first 5. Why "E" became "F" is another question, can't answer it, could speculate but don't want to get dinged.

Now, you can easily attack the premise that grades are distributed normally, and you can also attack the premise that grades are distributed reliably. But theoretically these seemed to be the key argument for this kind of division of grades.