Genesis: Eve

Genesis 2-4

Previously mentioned was John Walton's book The Lost World of Genesis One. Though he made several valid points about the role of temples in Near Eastern creation myths, Walton fails to convince of his theory that Genesis 1 has nothing to do with the 'material' creation of the universe. It is necessary to mention his equally influential follow-up, The Lost World of Adam and Eve, which has similar positives and negatives.

Walton suggests that Genesis 2 is a direct chronological sequel to Genesis 1. Whereas Genesis 1 portrays the creation of a well-populated humanity, Genesis 2 shows the first calling of a man named Adam from out of that existing population. Likewise, we do not read that God physically created the first woman out of the first man's rib; instead, Adam himself sees this within a symbolic vision, anticipating the intended biological purpose of men and woman as procreators. Thus, Genesis 2-3 portrays a historical event involving that literal man, woman, snake, and trees.

Walton struggles to convince the reader of his propositions. His attempts to appeal to other ancient Near Eastern myths to find precedent wind up backfiring completely; every text he cites to support his unusual interpretation ends up reinforcing the universal position that Genesis 2 indeed depicts the creation of the first man from the soil, and the first woman from the man's rib. That the texts he cites are plainly mythological only validates a mythological interpretation of Genesis 2-3.

Arthur George and Elena George indicate that, in the specific manner of Eve's creation (from one of the man's ribs), there is no immediate parallel in any ancient Near Eastern literature, with one exception.

one possible source of inspiration often mentioned by commentators is the Sumerian myth known as Enki and Ninhursag, discussed earlier. In that story Enki's rib ails him and the goddess that Ninhursbag sends to cure him is Ninti, who name means "Lady of the rib." But the Sumerian word ti as a verb also means "to make live," so Ninti also came to be known as the "Lady who makes live" or "Lady of life." In the myth, therefore, there was wordplay on "rib" and "life." This recalls Eve (in Hebrew ḥawwâ, meaning "life") both being created from a rib and being given the epithet, "mother of all the living" (Gen. 3:20)
George & George, The Mythology of Eden, 218

As with the wordplay of adam (man) designating his purpose to work the adamah (ground), so the woman is named havvah (lifegiver) to designate her purpose to give birth to hay (living thing).

tl;dr

The story of the first woman's creation from the first man's side is etiological. It is meant to convey the inherent procreative biology of a male and female union, and under the ancient Israelite understanding this union was meant to 'complete' the male and the female.