At first, it might be rocky. Tom would be suspicious of Harry's intentions. But as Harry shows him genuine care, Tom would start to open up. Eventually, they could become like family, with Tom looking up to Harry as a role model.
One possibility is that Harry, having seen the dark path Tom Riddle took, decides to take him in early. He might teach Tom about love and friendship, things he lacked growing up. Harry could introduce Tom to the wizarding world in a more positive light, showing him the joys of Hogwarts and the value of good relationships, thus preventing him from turning into Voldemort.
Perhaps in the fanfiction, Harry adopts Tom Riddle when he's still a young boy at the orphanage. Harry then uses his own experiences of growing up without parents to bond with Tom. He teaches Tom about magic in a different way than Dumbledore did, focusing on the positive aspects of magic for good. This could lead to Tom not becoming the evil Voldemort we know.
One common theme is redemption. Since Tom Riddle later becomes Voldemort, the idea of fem Harry being able to change his path and redeem him is often explored. It gives a sense of hope that even the darkest of characters can be saved.
As the story progresses, their relationship could become more complex. Harry might become a father figure to Tom, teaching him right from wrong. Tom, on the other hand, could start to see Harry as someone he can rely on, but also struggle with his own inner demons that might pull him away from Harry's influence.
Hermione would probably be very cautious at first. She would be worried about the potential danger that Tom Riddle could pose, even as a child. But she might also see it as an opportunity to study the roots of dark magic and how to prevent someone from turning to the dark side.
One popular 'fem Harry adopts Tom Riddle fanfiction' is 'A Different Kind of Family'. In this story, the female Harry takes Tom under her wing in a really interesting way. It explores how their relationship develops over time, with some unique twists on the magic world we know from the original series.
At first, it would probably be really tense. Tom is used to being on his own and having his own dark thoughts. Harry would be super cautious. But as time goes on, Harry's kindness might start to break through Tom's barriers. They could become like brothers, with Harry teaching Tom about the good in the world and Tom starting to trust Harry.
I really liked 'My Adopted Son, Tom Riddle'. This fanfiction has a unique take on the idea. The reader not only adopts Tom but also tries to shield him from the dark influences at Hogwarts. It has some great plot twists and the writing style is very immersive.
One possible plotline could be that Harry, having seen the destruction Tom Riddle caused as Voldemort, decides to take young Tom under his wing to prevent him from going down that dark path. He could introduce Tom to the love and friendship he never had, like the friendships Harry formed at Hogwarts. This new environment and positive influence might change Tom's view of the world.