Q – Doesn’t the naming of Eve show that Adam had authority over her?
A – No, both male and female were created in the image of God and they were given shared authority to rule (Gen 1:28).
There is a Jewish tradition that the one who does the naming of another has authority over the one named. There is, however, no evidence that Adam had authority over the Woman when he called her ‘Woman’ and later named her ‘Eve’.
The two occasions where Adam ‘named’ the Woman were unlike the naming of the animals that God brought ‘to the man to see what he would call them.’
The first time that Adam saw the woman he exclaimed, ‘she shall be called Woman’. This was done out of a response to what he saw and out of his free will and NOT by God’s command, leading, or for God’s own interest (as with the animals).
The second time, when he names her ‘Eve’, it is after the Fall and so we cannot interpret anything from this with regard to God’s original intention that is seen in Gen 1:28.