Isolation. Snape is a lonely character. He doesn't have many friends, and he is constantly in a difficult position between the Death Eaters and the Order of the Phoenix. His feeling of being alone and not being fully understood by anyone can lead to crying in fanfiction.
One possible reason could be the extreme stress he endures. In many fanfictions, Snape has to constantly balance his double - agent role during the war. The pressure of keeping his true loyalties hidden from Voldemort and his Death Eaters while also trying to protect the students at Hogwarts can be overwhelming, leading to a collapse.
One possible reason could be his self - loathing. He has a complex past and has made many difficult choices. In fanfiction, the authors might explore his inner turmoil through self - harm as a way of expressing his pain. He may feel that he doesn't deserve any better because of his actions and mistakes.
One reason could be that he realizes Harry has a lot of potential and wants to nurture it. He sees that Harry is in a difficult position, being the target of Voldemort, and decides to help him despite their past differences.
Perhaps she's being bullied. In the wizarding world, there could be other students who pick on her because she's a 'Mudblood'. This could be really distressing for Hermione, and when Snape finds her crying, he might be conflicted between his general dislike for Gryffindors and his sense of justice.
One possible reason could be remorse. Harry might realize that he misjudged Snape all along. Snape had a complex role, and if Harry discovers the truth about Snape's true loyalties and sacrifices, it could lead to him crying out of guilt for how he treated Snape in the past.
One possible reason could be that fanfiction often takes creative liberties that may distort the complex character of Snape that J.K. Rowling created. Snape has a very intricate backstory and personality in the original series, and fanfiction might simplify or misinterpret it.
One great Severus Snape cries fanfiction is 'The Unspoken Tears'. In this fic, it shows Snape crying after a particularly difficult encounter with the memories of Lily. The author does a wonderful job of delving into his complex emotions.
One reason could be to add a shock factor. It completely changes the expected family relations in the original story. Another reason might be to explore Snape's character more deeply. If Harry is his son, it gives a new perspective on Snape's actions and emotions towards Harry throughout the series.
He could also cry because of problems with his friendships. Maybe he has a falling out with his best friend over something silly but it really affects him deeply. In fanfiction, this is a common scenario that shows his emotional side.
One possible reason could be that Snape is secretly trying to protect Harry. Maybe the potion has some sort of protective charm or antidote against a dark magic threat that Snape has foreseen. He may not show it openly, but his actions are driven by his long - standing, complex feelings towards Harry's mother and his sense of duty.