Genesis 1:20 - "Now the man called his wife's name Eve, because she was the mother of all the living."

It is a small thing, but one that we should notice. 

Adam names Eve.  He was given direction to name all those with him in Genesis 2:19.  It is a fine thing that he continues this tradition here with his wife.

The reason we want to pay attention to this happening is the fall.  Man has just displayed his sin.  Adam and Eve are about to get kicked out of the garden.

This is the first display of authority after sin, and the last display of authority in the garden.  A simple thing like naming his wife is Adam's final show of desire for the life of old; a life where he can display the authority of God in daily life.

We all reach for that authority.  All of us desire to live the life of 'life', though death is what has been promised us.  We sin, we forfeit our connection to God, we desire connection with God...

Enter grace.

We see God allow this authority display to hold Adam close to his purpose.  Grace covers our sin and reminds us that God still has something for us:  life.