‘And God said to Abraham, “As for Sarai your wife, you will not call her name Sarai, but her name shall be Sarah (princess).”
The new name again stresses a new beginning in a new situation. Sarah is to share in Abraham's honour and her position as the producer of the chosen line is emphasised. She is given a new dignity and brought directly into the covenant, receiving a personal blessing. (Sarai is probably simply an older form of Sarah. It is the change of name and not the change of meaning that is significant).
“And I will bless her and give you a son from her. Yes, I will bless her and she shall be a mother (princess?) of nations, kings of peoples shall be of her.”
God declares that Sarah is to have a natural son in spite of her age, and that she too will have nations and kings who will look back to her as their source. There is no word for ‘mother' in the original, it has to be read in. On the basis of the name change and the context it may be that ‘princess' is the idea to be read in. Not just a mother but a mother-princess. The promise is of course an extension of the promise to Abraham.