Yes, they were "equal" before sin, so what gave "Adam" the right to "give his wife a different name"
××“× adam - man, or more appropriately "mankind."
you're conflating two stories, i think. one in genesis 2, one in genesis 3. genesis 2 was written to give the hebrew marriage custom a traditional origin. genesis 3 was written to explain things like agriculture, pain in childbirth, snakes, etc.
if you are refering the name in genesis 2, "woman," "man" as a grammatically equall name.
the bit about "called her woman, because she came from man" is simply establishing the origin of the grammatical relation between the masculine and feminine genders in the hebrew language. it's by coincidence that "man" and "woman" have three of the same letters in english, but the relationship is much more obvious in hebrew.
adam gives his wife a proper name only after the curse. the name he gives here is:
×—×•×” chavah - "causes life."
and "rule over her?"
this is part of god's curse. it was probably written to explain the origin of patriarchal society, and the ancient hebrews often deeply misogynistic attitudes. they were equal before that point.
If they were equal, then "Eve" should still have as much rule over "Adam" as he does over her because they both "sinned" and aren't all sins equally egregious under the eyes of God?
each guilty party was given different curses. eve took charge and led her husband astray -- so her curse is to follow him. adam violate god's garden -- so he has to make his own. and the serpent spoke in contradiction to god -- so he has to eat dirt.
anyhow. really, to use the text as justification is circular, since the text was written to explain practices contemporary to the authors. all people are saying by quoting the bible as justification on this issue is "let's live in ancient israel."