A couple of myths:
The term "pure functional" doesn't come from mathematics, where all functions are by nature "pure" and, so, there was never any need to call anything a "pure function".
The term doesn't come from imperative programming. The early imperative programming languages, Fortran, Algol 60, Pascal etc., always had two kinds of abstractions: "functions" that produced results based on their inputs and "procedures" which took some inputs and did an action. It was considered good programming practice for "functions" not to have side effects. There was no need for them to have side effects because one could always use procedures instead.
So, where else could the term "pure functional" have come from? The answer is - sort of- obvious. It came from impure functional programming languages, the foremost among them being Lisp. Lisp was designed sometime between 1958 and 1960 (between the first and second reports of Algol 60, whose design McCarthy was involved in, but didn't feel satisfied with). Lisp's design was based fundamentally on functional programming. However, it also allowed side-effects as a pragmatic choice. It did not have a notion of a command or a procedure. So, in Lisp, one mostly wrote "pure functions", but occasionally, one wrote "impure functions," i.e., functions with side-effects, to get something done. The terms "pure Lisp" or "purely functional subset of Lisp" have been in use for a long time. Slowly, by osmosis, this idea of "purity" has come to invade all our space.
The imperative programming languages could have resisted the trend. But, once C decided to abolish the idea of "procedures" and call them "void functions" instead, they didn't have much of a leg to stand on.
