Well, it might be a story where a child shares something special about their big brother with their mom. For example, the child could be sharing the big brother's achievements, like he won a school competition and the child is excitedly telling mom all about it. Or it could be about sharing a secret that the big brother told the child, which the child then decides to tell mom.
I'm not sure specifically what this 'Sharing Big Brother with Mom' story is without more context. It could be about sharing an experience or possession related to an older brother with one's mother.
Perhaps it's a story about sharing a relationship. Maybe the big brother has been very kind and helpful to the child, and the child wants to share this wonderful relationship with their mom. It could also be a story set in a family situation where resources are limited, and the child has to share their big brother's things with their mom, like sharing a book that the big brother loves and has passed on to the child, and now the child is sharing the story in the book with mom.