Tell the bully directly that it's not acceptable to use your mom's stories in a mean way. Be assertive and show that you won't tolerate it.
First, you should talk to your mom about this situation. Let her know how you feel and that you don't like the bully having those stories. Then, if possible, you could confront the bully together with your mom and firmly ask the bully to stop spreading or using those stories.
Try to change the subject when she starts to tell the same story. For instance, if she begins, you can quickly say 'Mom, did you hear about this new thing that happened at school?' This will divert her attention and stop her from going into the familiar story. Another option is to introduce her to new stories. Maybe give her a book of short stories and ask her to read them so she has new material.
If a direct approach doesn't work, you can try to limit his access to opportunities to share these stories. Avoid conversations where he might bring it up. If he still manages to, quickly change the topic. You can also create a situation where it becomes clear that you are not interested in hearing or having others hear these stories. For instance, you could say in a group setting (if he shares there) that you'd rather keep your family life private and not have it be the topic of discussion at work.
One way is to make your stories really boring. For example, if you just post about the same mundane things every day without any new or interesting angle. But this might also affect other people who do enjoy your stories. Another option is to directly tell me that you don't want me to watch your stories. It's a bit blunt but it gets the message across.
You can offer her a special treat if she sleeps in her own bed for a certain number of nights. For example, a small toy or a piece of her favorite candy.
One possible interpretation is that the bully has some sort of psychological hold over the mom. Maybe through threats or intimidation towards the child, the mom feels she has to comply with the bully's unspoken demands. For example, if the bully says he'll harm the child if the mom doesn't let him have his way in some small matter.
Tell him that bullying is not acceptable. Just be straightforward and firm about it.
Another way to look at it could be in a neighborhood setting. The bully is a troublemaker in the neighborhood. He might do things that make the mom feel pressured or restricted. For instance, he might play loud music all night which annoys the mom but she feels powerless to stop him, as if he has some sort of 'ownership' over her peace of mind.
If it's someone you know, you can directly tell them it's unacceptable and a violation of privacy. Threaten legal action if necessary.
You could try changing the content of your stories. Make them more interesting, for example, by sharing some exciting experiences or new things you've learned.