The title is reminiscent of the movement among certain Christians. Name it and claim it. I’m not going to break down the theology of this movement, but address the theology of the title and its implications for Genesis.
Name it: the power to name a thing makes one the maker or owner of the thing
Claim it: based on the power to name the thing, I can claim ownership or Lordship over it.
I have a daughter. My wife and I have named her Abrielle Joy O’Rourke. We have the right to name her, because we were given the power to create her. We are her parents/ lords, we rule her life.
It has historically been thought that man named woman, as it has been believed that man named the animals. If man named the animals, that would make him their maker and lord. If man named woman, he would be her maker and lord/father at best, but not her husband. This is why the angel of the Lord could give Jacob the new name of Israel (Genesis 32:28), but refused to give his name to Jacob (Genesis 32:29). To give a name is to designate character, thus the angel changed Jacob's name from “heel-grabber” to “prince of God”.
So who named woman?
According to Genesis 2, God, looking at Themself, and then looking at man single, and wanting to preemptively mitigate any desire for sin, decided to make man as much like Himselves as possible. God decided to make man singular and plural, gendered, married, and intimate. Howbeit God did not tell man what They were planning. God simply decided to test man:
Gen 2:19 And out of the ground the LORD God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.
Gen 2:20 And Adam gave (called) names to all cattle, and to the fowl of the air, and to every beast of the field; but for Adam there was not found an help meet for him.
Some have taken this to mean that God tried to make Adam a mate from the animals first, to see if Adam would chose one of them as his equal. This is unnecessary. The Bible says that whatever Adam called the animals, that was the name (thereof or in fact). The idea is not that Adam invented names for the animals, but rather that God was testing the image to see if it knew what God knew. So God brings Adam a dog, Adam looks and says, "thats a dog", and God says, "you are correct; it is a dog". Adam then discovers again something that God already knew: there was no one like him. Then God made woman.
Gen 2:23 And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man.Adam did not name woman. Adam look at her and exclaimed that she was him, “bone of my bone and flesh of my flesh”. J Doukhan and R. Davidson have rightly understood that it is God who gives woman her name: "She shall be called woman for she was taken out of man" Adam can call the name of the animals because they had names before he met them, and God simply tested him to see if he knew what They knew. However, woman received her designation from God, and it is pronounced in verse 23. Adam can only exclaim his excitement at seeing his equal. However he is not her Lord nor Creator, therefore he has no right to name her. That is the prerogative of the Divine.