The baby could be the child of a minor character from the original series who was in hiding. For example, the baby might be the offspring of a wizard who was on the run from Voldemort's followers and unfortunately didn't make it, leaving the baby behind for Harry to find.
Maybe the baby is a result of some sort of magical experiment gone wrong. And Harry Potter finds it in a hidden place where the experiment was carried out. The baby might have some strange and unpredictable magical abilities as a consequence.
Well, perhaps in this fanfiction, Harry Potter stumbles upon a baby while on one of his adventures. Maybe the baby has some sort of magical power that he has to protect from the dark forces. It could also be that the baby is related to an old prophecy and Harry has to figure out how to keep it safe and fulfill the prophecy at the same time.
In fanfiction, if Harry Potter finds a baby phoenix, it could be a very magical encounter. The baby phoenix might be in need of protection. Harry, being the hero he is, would likely take it under his wing. He could use his knowledge of magical creatures from Hogwarts to care for it. Maybe the phoenix would grow up to be a powerful ally for Harry in his future adventures.
One popular one is 'A Different Destiny' where Severus finding baby Harry sets off a chain of events that deviate from the original story's path. It has great character development.
In the 'harry finds a baby fanfiction', there are numerous possibilities. It could be set in a world where Harry is a grown - up and he stumbles upon a baby that is crucial to the future of his world. The story might revolve around how he deals with this newfound situation. He could enlist the help of his friends to take care of the baby. There could also be a mystery surrounding the baby's origin that Harry is determined to solve, which could lead to all sorts of adventures and revelations.
In this kind of fanfiction scenario, Snape might initially be conflicted. On one hand, he has his long - standing dislike for James Potter, Harry's father. But on the other, his sense of duty as a Hogwarts professor could kick in. He might reluctantly start to heal Harry, all the while muttering under his breath about how much trouble Harry always seems to get into, just like his father. Then, as he heals Harry, he might start to see some of Lily, Harry's mother, in him and soften a bit.
There could be. Fanfiction writers are known for their ability to combine the most unexpected elements. Samehada is a very unique element from the Naruto universe and Harry Potter has its own rich world. Some writers might have been inspired to create a story where Harry Potter somehow finds Samehada. You could start your search by using very specific search terms on fanfiction search engines. If it exists, this kind of fanfiction would likely be in the crossover section of the relevant platforms.
One interesting fanfiction could be about baby Harry's first magical experiences even before he went to Hogwarts. Maybe he unknowingly made his toys levitate when he was really young, and his aunt Petunia was terrified but couldn't figure out what was really going on.
His best friends Hermione and Ron would surely be there to help. Hermione with her vast knowledge of magic could research about the egg, and Ron with his loyalty would stand by Harry's side in any danger related to the egg.