MISUSING A CUSTOM
There is no evidence in both the incidents of Adam “naming” Eve that God had given him authority over her. I say this to counter the willy-nilly correlating of these incidents with a Hebrew custom that if you named something it indicated that you had authority over it.
If it was a Biblical custom, then its origins may lie in Adam naming the animals. However, the naming of the animals was only shown to be out of God’s interest to see what Adam WOULD name them. Yet, even if it did imply authority over the animals, this incident was by God’s leading, whereas the “naming” of his counterpart was not.
ADAM CALLED HER “WOMAN” AND NAMED HER “EVE”
Note, that the first time that Adam saw the woman, he exclaimed out of his free will and NOT by God’s command, leading or for His interest (as with the animals), “she shall be called Woman.” By doing this, he did nothing wrong, but neither was he under God’s direction. He was simply responding to having seen His counterpart.
Then, later, when Adam names the woman, “Eve”, in Genesis 3:20, it is after the fall and God did not command this either (though He may have inspired the choice of name). So, once again, this could be simply Adam’s free will (perhaps with God’s inspiration), but because it is after the fall, it could also be argued that this was Adam’s first act of ruling Eve as Genesis 3:16 said would happen.
There is therefore no evidence that by Adam calling the person made from his side “Woman” and later naming her “Eve”, that these acts of naming her had any connection to having authority over her.
In an effort to prove their point that Adam had authority over Eve, some have argued that slaves, children, and animals were named by those who were over them. This was true, but slaves, children, and animals don’t equate to the Counter-part Woman given to Adam. She was not a slave, child or animal, but that teaching produces the fruit of people treating women this way.
Rather than be subject to him, the Woman was to be a strength alongside Adam (his Ezer Kenegdo – a help as his counterpart). She would be strong where he was weak and together they would have co-dominion over every living thing, with the mission to subdue the earth (see Gen 1:28).
(Note: Ezer is often used of God helping us. Kenegdo shows the help is of an equal. From word study of “help meet”, EZER KENEGDO, God’s Word to Women).