The child could be an orphan from a small, destroyed village that Naruto comes across during one of his missions. This child might have lost their family to a powerful enemy and Naruto feels a sense of responsibility to take them in.
It could be an orphaned wizard child from a family that was killed by Death Eaters. This would give Snape an added sense of responsibility as he might feel guilty for being associated with the Death Eaters in the past.
In this fanfiction concept, Ash's adoption could be a very heartwarming story. Maybe he meets an orphaned child during his travels. Ash, with his kind heart, decides to take the child under his wing. It could be a chance for him to pass on his knowledge of Pokemon and values like friendship and perseverance.
One idea could be that Percy discovers a demigod child who has been abandoned in the mortal world. He decides to adopt the child to protect them from monsters and to give them a proper home at Camp Half - Blood. He then has to teach the child about their demigod heritage and how to use their powers.
Perhaps the child is the offspring of a Titan. This would make for an interesting story as Percy would have to deal with the potential danger and mistrust from other demigods. But Percy, being Percy, would see the good in the child and adopt them regardless. The child might have unique powers that set them apart from other demigods, like the ability to control time or space.
He could adopt a child who has been ostracized by the village for some reason. Naruto, who has always been an outcast in a way, would see himself in the child. This type of adoption would allow for a story about acceptance and redemption. Naruto would strive to show the village that this child is worthy of love and respect, just like he once had to prove his own worth.
The plot might be that Sakura has always wanted a family but couldn't have one in the normal way. One day, she comes across a child who has been abandoned. She instantly feels a connection and decides to adopt. Along the way, they bond over shared interests. For example, if Sakura likes gardening, she teaches the child about plants and they grow a beautiful garden together, strengthening their relationship.