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.
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.
There could be an original character created just for the fanfiction. A woman with a kind heart but a tough exterior, who was a great fighter. She could have met Luffy's father during a battle and they fell in love, and she became Luffy's mother.
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.
Happy might also become Luffy's friend. Luffy would be amused by Happy's flying ability and his cute appearance. Happy could also help Luffy navigate the Fairy Tail world, as he knows it well. And Luffy might share his food with Happy, which would make Happy very happy indeed.
She could be a former pirate captain herself. Maybe she was one of the most feared pirates in her time, and her skills and reputation were passed down to Luffy in some way, even if he didn't know it was from his mother.
Rebecca could potentially be in Luffy's harem. After Luffy helped her in Dressrosa, she might develop feelings for him. She is a strong - willed and kind - hearted character. Another could be Shirahoshi. Luffy has a special connection with her as he saved her from being captured. Her gentle and innocent nature would contrast with some of the other more assertive members of the harem.
Aside from Luffy, characters like Ace might show up in this type of fanfiction. Ace, being Luffy's brother, could have some connection to Luffy's discovery of conquerer's haki, perhaps through memories or some form of spiritual guidance. Additionally, some powerful enemies might be present. These enemies would serve as the motivation for Luffy to quickly master the conqueror's haki in order to defeat them and protect his crew. There may also be some minor characters from the One Piece world who assist Luffy in his learning process, like the inhabitants of a special village that has knowledge about haki.