As I understand it, there was no plan for breeding, therefore there was no contract or agreement to use your mom's dog as the stud. So the truth of the matter is that your mom is not
entitled to a "pick of the litter" stud fee. That is not to say that you can't give her one if you choose to. If my mom ever wanted one of my puppies, she would definitely get one, just because she is my mom.
