In Genesis 2, Adam names the animals. To name something is to claim authority over it, and earlier (Gen. 1:28) God placed everything that moves on the earth under the rule of those made in His image.
But Adam doesn’t bother to name his wife…until after the Fall. The man composes a poem, but doesn’t take time to compose a name.
After God lays down his curse, part of which is that the man will rule over the woman (Gen. 3:16), only then does Adam name Eve. A lot of things changed as a result of the Fall. Could one of them have been the nature of the man’s authority over his wife? Adam now claims his right to rule his wife by naming her, as he did the animals.
Paul makes it clear that the created order itself has some authority, and it included some form of hierarchy. Man is the image and glory of God, but woman is the glory of man (1 Cor. 11:7). Adam was created first, and then Eve (1 Tim. 2:13).
Likewise, the whole point of marriage, from the very beginning, was to image Christ and His Church, a clear relationship of ruler and subject. Pre-fall, the authority was in place, but the placement of the naming of Eve hints that something truly precious was lost between the sexes, a corrupting of the authority that lowered the woman closer to the status of a animal, less of the ideal helper than intended, and as a result both man and woman are diminished.