One possible reason could be that Obito sees Naruto as a means to fulfill his own twisted ideals. Maybe he wants to shape Naruto into a weapon against the Hidden Leaf. Another reason might be that he feels some sort of connection to Naruto because of their shared experiences of being alone and ostracized in the past. He could also think that by taking Naruto as a baby, he can prevent Naruto from becoming the hero the village wants him to be, thus disrupting the balance in the ninja world.
The story could develop with Obito training Naruto in his own way, away from the values of the Hidden Leaf. Naruto might grow up with a different set of skills and a different view of the ninja world. There could be a lot of internal conflict in Naruto as he discovers the truth about his origins and the fact that he was taken from his village. This could lead to a showdown between Naruto and the Hidden Leaf at some point, or Naruto might try to find a middle ground between Obito's teachings and his own sense of right and wrong.
Obito might be initially hesitant and a bit awkward around Baby Naruto. But as time passes, he could develop a soft spot for the baby. He might play simple games with Naruto like peek - a - boo or make funny faces to make him laugh.
In this fanfiction, Obito's development can be quite interesting. He may begin with his own motives that are not entirely pure. However, as he spends more time with Naruto, he starts to see the world through Naruto's eyes. This could lead to him becoming more empathetic towards others in the ninja world. For example, he might start to understand the importance of protecting the village and its people, just like Naruto does.
One unique plot point could be Obito teaching Naruto some of his own special jutsu that he developed during his time as a rogue ninja. Another might be Obito using his knowledge of the darker side of the ninja world to protect Naruto from certain threats that the normal Konoha ninja might not be aware of.
Naruto uses his ninja skills. He's always on high alert, using his enhanced senses to detect any potential danger. His quick reflexes mean he can react in an instant if anything threatens the baby.
In a fanfiction, Obito might be sent on a mission by some unknown force to watch over Baby Naruto. As he does, he starts to question his own beliefs. Naruto's pure and unwavering will even as a baby might start to chip away at Obito's hardened heart. This could lead to Obito trying to influence Naruto's growth in a positive way, like teaching him some skills secretly without Naruto knowing it was Obito.
In the fanfiction, Naruto would be super protective. He'd use his ninja skills to keep the baby safe from any harm. For example, if there was a threat, he could quickly create a barrier around the baby using his chakra. Also, he'd be really patient, which is a bit different from his usual impulsive self when it comes to taking care of the baby.
Obito's influence would likely make Naruto more aware of the darker side of the ninja world. Naruto is usually very optimistic, but with Obito's guidance, he would see the hardships and betrayals that can happen. For example, Naruto might become more cautious in his trust of others.
One possible fanfiction could be about Naruto and Obito teaming up to face a new, powerful enemy. They could combine their unique skills, like Naruto's Rasengan and Obito's Kamui. Their relationship could start off rocky due to their past differences but gradually turn into a strong bond as they fight side by side.