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.
Nami could also be a common character. She might help Luffy manage the financial aspects of his admiral duties, as well as use her navigation skills to assist in any navy - related voyages. And of course, she'd be there to keep Luffy in check when he gets too carried away.
Rayleigh would be a great candidate. He's an experienced pirate and has already trained Luffy for a while. He has a lot of knowledge about the world of pirates and could give Luffy a talk about the deeper secrets of the Grand Line or how to deal with the powerful enemies that lie ahead.
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.
He could face powerful sword - using pirates. Since Luffy now has a sword, he would attract the attention of other sword - masters in the pirate world. These opponents would test his new - found sword skills and force him to improve quickly.
Some fanfictions also pair Luffy with Hancock. Given Hancock's infatuation with Luffy in the original series, it's not a stretch to develop a full - blown love story. Her powerful status as a pirate empress and Luffy's carefree nature can create an interesting dynamic, with Hancock trying to win Luffy's heart while he is mostly oblivious to her romantic intentions.