One possible plot could be that Harry discovers a young Tom in a difficult situation, perhaps an orphanage similar to his own past. He decides to take Tom under his wing out of a sense of empathy. Harry then tries to guide Tom away from the dark path he originally took in the canon. For example, he might introduce Tom to the world of positive magic and friendship early on, which could change Tom's perspective on life and magic.
You can start by looking on Archive of Our Own (AO3). It has a large collection of fanfics, and you can use the search function to type in 'Harry adopts a Tom' to find relevant stories.
One popular 'Harry adopts a Tom fanfic' is 'A New Beginning'. In this fic, Harry discovers a young Tom in a difficult situation and decides to take him in. It explores how their relationship develops over time, with Harry trying to guide Tom away from the dark path he was destined for in the original story.
One common theme is redemption. Since Tom Riddle becomes Voldemort in the original story, these fics often explore the idea of whether he can be redeemed with Harry's help. Another theme is the power of family. Harry, having grown up without a proper family for a long time, tries to create a loving family for Tom. And the theme of changing destiny also runs through. It's about whether they can change the dark future that was set for Tom in the original story.
Hermione would likely be a main supporting character. She could help Harry in coming up with ways to educate Tom and keep him on the right path. Also, Ron might be there, providing some comic relief and also acting as a sort of brother figure to Tom, showing him what a normal family - like friendship could be like.
There's also 'Redemption's Chance'. This fanfic focuses on Harry taking Tom under his wing. It shows Harry using his own experiences of loss and struggle to connect with Tom. The story is filled with heart - warming moments as Harry tries to give Tom a loving home environment. It also has some great magical elements and battles that are unique to this new take on the Harry Potter universe.
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.
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.
In this kind of fanfiction, perhaps Severus Snape and Tom (presumably Voldemort in his Tom Riddle form) would have a very different dynamic with Harry. Maybe they would try to shape Harry's magical abilities in their own ways. Snape might use his knowledge of potions and dark magic to train Harry, while Tom could introduce Harry to some of the more complex and dangerous aspects of magic from a different perspective.
One possible plotline could be Harry seeing the potential for good in Tom even though he knows of Voldemort's future. He might try to raise Tom with love and kindness to change his path. For example, Harry could introduce Tom to the wizarding world in a more positive light, showing him the beauty of magic for good rather than for power.