Another way the story could go is that the baby has a unique ability that attracts the attention of other villages. Naruto then has to protect the baby from those who might want to use its power for bad. He could train the baby to control its power and also form alliances with other villages to ensure the baby's safety. Along the way, the baby could learn about Naruto's own past and be inspired to become a strong and kind ninja like him.
In this fanfiction, Naruto might take the baby in and raise it as his own. He could teach the baby ninjutsu skills he has learned over the years. Maybe the baby has some special powers that are gradually discovered as the story progresses.
In a fanfiction where Orochimaru kidnaps baby Naruto, the development could be very complex. Orochimaru could initially plan to use Naruto's Uzumaki bloodline for some powerful jutsu creation. But as Naruto grows, he might show kindness towards Orochimaru's minions, which could start to change the dynamic in Orochimaru's hideout. Back in the village, the absence of Naruto could lead to a shift in power. Some might try to take advantage of the situation while others would be focused on getting Naruto back. Naruto himself could end up becoming a sort of bridge between Orochimaru's world and the Hidden Leaf Village if he manages to escape or change Orochimaru's ways.
Naruto's character could develop morally if he finds Mjolnir. Since Mjolnir is only liftable by those who are worthy, it would reinforce Naruto's sense of justice. He'd see himself as even more of a protector. He might start to view his ninja way in a new light, perhaps incorporating some of the values associated with Mjolnir. For example, he could become more strict about who is worthy of his protection, like Thor is with Mjolnir.
The baby could turn out to have some sort of special chakra. Naruto, being curious and always wanting to help, tries to find out more about it. He might seek help from Kakashi or other experienced ninjas.
Well, Naruto might try to protect the baby dragon from any harm. He could use his ninja skills to keep it safe from enemies who might want to capture it for its power or rarity.
Tsunade's character often develops a stronger sense of purpose. In fanfictions where she finds baby Naruto, she realizes that she has a new mission, which is to raise Naruto to be a great ninja. This might lead to her changing her ways in terms of leadership in the village. She might start to focus more on the future generation and how to make the village a better place for Naruto and others to grow up in. She also may become more empathetic towards others in the village as she experiences the joys and challenges of raising Naruto.
The story could develop with Naruto being stolen by a group of dark wizards. They plan to use his unique energy for their evil schemes. However, Naruto is rescued by a group of good wizards who sense his pure heart. As he grows up among them, he becomes a symbol of hope in the wizard world. He then uses his ninja skills, which he unconsciously has, to fight against the dark wizards who once tried to use him. His journey could involve learning more about the connection between the wizard and ninja worlds.
One idea could be that the baby is from a hidden clan. Naruto decides to protect the baby from those who would use its special abilities. He takes the baby under his wing and trains it as it grows up, passing on his ninja values.
The baby could be an alien species that Goku has never encountered before. It might have special powers related to its alien origin, like the ability to control energy in a different way than Goku and his friends.
The baby could be an ordinary human baby that has some sort of latent magical ability that Raven senses. She then decides to protect the baby from the human world that may not understand or accept the baby's potential magic, and also from the magical world that might want to exploit the baby.