The best place to find information about much of Mesopotamian culture is through Hammurabi's Code. Hammurabi was a king in Mesopotamia who is most famous for creating a very strict and thorough code of laws for his empire. Included in the code are many references to women and children which provide us with a clear picture of their status. Mesopotamia was a very patriarchal society in which men literally owned their wives and children. This is clear through Hammurabi's Code where it is shown in laws 170-180. An example of this is that men could literally sell their wives and children into slavery to cover debt.
While there is patriarchy in the laws of Mesopotamia and men controlled the lives of their women and children, this is one of the first societies where women did retain some rights. As you read through the laws pertaining to family relationships you will notice that women were allowed to retain their dowry in the event of the death of their husbands. Children, even illegitimate children, were given rights of inheritance based on a variety of different situations.
The family relationships defined in Hammurabi's Code show the status of women and children in Mesopotamia. I suggest that after reading this you take some time to check out the text of the code for further examples. I will include a link to the full text with this answer!