Another storyline could be that the Akatsuki, seeing baby Naruto as a valuable asset, try to raise him as one of their own. But Naruto's innate sense of justice, which is there even as a baby, starts to change the Akatsuki from within. He might influence them to question their more villainous actions and motives.
Pain could also be a major character. His ideology and the Akatsuki's goals under his leadership might be affected by the presence of baby Naruto. He might see Naruto as a potential threat to his plans or as an opportunity to change the Akatsuki's path. Also, Konan might be involved as she often has a more empathetic side compared to some of the other Akatsuki members and could play a role in taking care of the baby.
There are so many fanfiction stories out there, and 'akatsuki find a baby' is a unique enough idea that it could have been explored in various ways. For example, in some fanfictions, the baby could be a key to some hidden power or prophecy that the Akatsuki is interested in. Or perhaps the baby is from an enemy clan, and the Akatsuki has to decide what to do with it. To find the exact fanfiction you're looking for, you could try searching with specific keywords related to the Akatsuki members involved, or any other details you might remember about the story.
Pein could be another one. As the leader of the Akatsuki, he might see baby Naruto as a means to an end. He could either want to use Naruto for his own grand plans or, in a more interesting twist, be swayed by Naruto's innocence and change his perspective on the Akatsuki's goals.
One idea could be the Akatsuki members having to protect baby Naruto for some mysterious reason. Maybe they were tasked by a powerful entity to keep him safe from other dangerous forces in the Naruto universe.
Well, in this fanfiction, it could be that Akatsuki has a change of heart. Some members might realize that Naruto represents a hope for the ninja world in a different way than they initially thought. So they step in to save him when he's in a really tough situation, like being trapped in a powerful jutsu or surrounded by a large number of enemy ninjas. They use their own unique skills, like Kisame's water jutsu or Itachi's Sharingan abilities to fight off the threats.
One possible place to find such a fanfiction is Archive of Our Own. They have a large collection of Naruto - related fanfictions, and this specific plot might be there. Just use the search bar and type in 'akatsuki find a baby with special powers' and see what comes up.
One key element is the contrast between Sakura's innocence as a baby or young character and the often dark and mysterious nature of the Akatsuki. It creates an interesting dynamic.
In this fanfiction, the Akatsuki might first be really confused about finding a baby. Some members could be immediately protective, like Itachi. He might see the baby as an innocent life to safeguard. Others, like Hidan, might be annoyed at first but gradually grow fond. They could end up raising the baby within their hideout, teaching it their unique skills in a strange way, like how to move stealthily.
One common theme is responsibility. The Akatsuki members, who are usually seen as tough and somewhat aloof, suddenly have to face the huge responsibility of being a father.