This seems to be a hot topic right now on the forum.
I wish to say at the outset that I am using a definition of the word "idiom" similar to what is found in the American Heritage Dictionary: "a traditional way of saying something." This dictionary goes on to say that "often" an idiom will "not seem to make sense if taken literally." Often, but not always. I mention this because online one can find definitions of "idiom" which insist that the phrase must be figurative; then these same sites will turn around and list as idioms phrases such as "a man of few words" (which is so literal that a female will call herself "a woman of few words" instead) or "alive and well" ("alive" literally means alive here, and "well" literally means well).
"Born of a woman" obviously has a very literal element to it insofar as all humans literally have mothers who literally gave birth to them. However, it is anything but the most direct way to say "human being" (or "mortal"), since where the idiom is used most purely as an idiom the mother has nothing to do with the situation in context.
That said, here are a few instances of "born of a woman" used idiomatically. Since the sticking point for some people seems to be the exact word used in Greek to render what must have originated as a Hebrew turn of phrase, I will include the relevant term in the original language(s).
As might be expected, the Hebrew word used is completely consistent; it is always yalad. Also, the Hebrew parallelismus membrorum sets "one born of a woman" in parallel with "mortal" or "human" (both אדם and אנוש are used here). Translating the concept into Greek, however, produced several possibilities. Job (LXX), Matthew, Luke, and the Clementine Homilies all use γεννητός (the adjectival form of γεννάω); Sirach uses γέννημα (a noun related to γεννάω); Josephus uses γονή (a noun derived from γίγνομαι); and Origen uses γενόμενος (the participle of γίγνομαι, quite possibly under the influence of Galatians 4.4 itself, though the context does not demand this).
So some form of γεννάω is used most often, but some form of γίγνομαι may also be used sometimes. This lines up with the usage of γεννάω and γίγνομαι to indicate birth overall, outside of any idiom; the former is more common, but the latter is also used. Both are used to translate the Hebrew word ילד (yalad) in the LXX. Genesis 4.18 is an interesting case, because it has four instances of the Hebrew ילד, one of which is translated by the Greek γίγνομαι, the other three of which are translated by the Greek γεννάω, all in the same sentence (at least in the critical editions; I have mentioned before that the LXX manuscripts often differ within the same verse on which Greek word is used to translate yalad). There are other instances (Genesis 17.17; 21.3; 36.5) in which the sentence has two instances of the Hebrew ילד, one of which is rendered by γίγνομαι, the other of which is rendered by τίκτω ("to have a child").
I hope this is something close to what you were seeking.