Well, in the 'big mom is luffy's mom fanfic', they might have crafted a story where Big Mom is Luffy's mom. But it goes against what we know from the official 'One Piece' story. Fanfictions like this can be really interesting as they can rewrite the story in unexpected ways. They might have some interesting plot twists and explanations within the fanfic to justify this relationship, but it's not part of the main 'One Piece' universe.
One possible plot implication could be a change in Luffy's power source. Since Big Mom has such unique and powerful abilities, Luffy might inherit or be related to some of those powers in the fanfic. Maybe his Devil Fruit ability could be related to Big Mom's in some way.
As of now, in the manga, the identity of Luffy's mom remains unknown. There have been many theories and speculations among fans, but no official confirmation.
A fanfiction could depict Portgas D Rouge as a key figure in a larger story involving Sister Dragon and Luffy's mom. Rouge's sacrifice could have been related to something that Sister Dragon was aware of. Luffy's mom, on the other hand, might have passed on certain powers or knowledge to Rouge before her death. This could lead to a series of events in the fanfiction where Luffy discovers the truth about his mother through the memories and legacies left by Rouge and Sister Dragon.
The first step should be to encourage the mom not to give in to the blackmailer. Then, they should report the matter to the authorities, like the police. Blackmail is a serious offense, and the law can protect the victim.
If such a situation occurs, the mother should immediately seek help. She can call the police or contact local social services. Her safety is of utmost importance.
You need to protect your mom. Let her know she doesn't have to comply with the blackmailer's demands. Try to figure out who the blackmailer is. If it's someone you know, you might try to have a conversation with them to stop this unethical behavior. If that doesn't work, seek legal help.