The threat to his friends. James is very loyal to his friends, like Sirius, Remus, and Peter. If they were in danger or if something bad happened to them, it would be understandable for him to cry. For example, if one of them got seriously injured or was in a life - threatening situation, his emotions would be on full display.
One possible reason could be that he was poisoned. In the wizarding world, there are many dark spells and potions that could cause such a reaction. Another reason might be extreme motion sickness if he was on a wild magical ride, like on a particularly unruly broomstick or a flying magical creature.
One possible reason could be a dark magic influence on James Potter in the fanfiction. Maybe he's been cursed or possessed, causing him to act out of character and disown Harry.
It could be an exploration of the darker side of human nature. Writers may use these characters to show how relationships can go horribly wrong, even if it's not in line with the original story.
One reason could be for the sake of novelty. Fanfic writers like to create unique and unexpected scenarios, and this is a very different take on the Potter family. It grabs readers' attention.
Some fans might create it as a form of shock value. They want to stand out by presenting a very different and controversial take on the relationship between James Potter and Harry.
One reason could be his regret over his past actions. For example, when he distanced himself from his family to pursue his career at the Ministry. He might realize later how wrong he was and cry out of remorse.
One reason could be the loss of his father. In the Naruto universe, Kakashi had a complex relationship with his father's death. If a fanfiction delves deeper into his younger years, this could be a very emotional trigger for him to cry.
Guilt. For example, if he fails to save someone he cares about. He'll feel responsible and that could lead to him crying.
Perhaps James had some hidden magical ability that he was unaware of until that moment. It could be something that was passed down through his family but had never manifested before. This ability could have protected him from Voldemort's killing curse. Or, it's possible that there was a plot by some other characters to save James. There could have been a group of wizards or witches who had been secretly working to keep James alive because they knew his importance in the fight against Voldemort. They might have used some sort of time - turning magic or a protective charm that Voldemort couldn't break through.
Perhaps Snape believes that James has some crucial information that he needs and he thinks torture is the quickest way to get it. Or maybe there's a power struggle going on. Snape wants to assert his dominance over James, who might be seen as a rival in some way, either for a position at Hogwarts or in the wizarding community. Also, it could be due to a misunderstanding that has spiraled out of control, leading Snape to think that James is a greater threat than he actually is.