How did Adam and Eve get their names?

The name Adam means son of red earth in Hebrew. It can also be argued from the Bible that Adam means man.  When God made Adam in Genesis 2:7 He called him man and continued to call him this until Genesis 2:19 where he is called Adam.  Since the Bible uses both words to refer to the same person, we know that the name Adam means man and that God was the One Who named him so.  When God formed Eve, He never gave her a name.  After Adam woke up, he saw Eve and called her woman, meaning coming from man.  She is called woman from her creation in Genesis 2:22 until Genesis 3:20 where Adam changes her name to Eve, meaning mother of all living.  So, we see that both Adam and Eve also have the names man and woman, the former are used to identify the first two people God created while the latter refer to male and females in general.