The name, we might assume, was an honorific and one which carried with it power and hence the respect of the Pharaoh's subjects. Such a gift portends a bright future. The other gift, a woman, promises that that future will be filled with not only the honor and glory of power but also the humility and tenderness required to live at peace with a woman. And children, for Asenath was fertile and provided for Joseph Ephraim and Manasseh.
It is unwise, though, for a man to reject gifts from the King. And this is so when dealing with mortal kings, but how much more must one avoid the rejection of gifts from the Heavenly King.
To him that overcometh will I give to eat of the hidden manna, and will give him a white stone, and in the stone a new name written, which no man knoweth saving he that receiveth it. --Revelation 2:17b (KJV)