It could also be about Harry's future. Snape might be giving advice on how to deal with Voldemort. He could be sharing his insights on the Dark Arts and how Harry can use that knowledge to his advantage without being corrupted by it. Snape has a complex understanding of the Dark Arts and might be trying to prepare Harry for the battles ahead.
Perhaps Snape gives them the talk about the importance of unity in the face of the dark forces that are always lurking. He could be using his own experiences to make them understand that they should put aside their differences for the greater good. For example, he might talk about how during the war, internal disputes among wizards only made it easier for Voldemort to gain power. By uniting, they could be a stronger force against any potential threats. Also, Snape might use this talk to make them realize that their personal grudges are petty compared to the bigger issues at hand.
Maybe they would talk about Voldemort. Since Nagini was Voldemort's pet and weapon, Harry might ask her about Voldemort's plans or weaknesses.
It could be a sign of an unexpected alliance. Maybe Harry realizes that Snape has some hidden motives that are actually for the good, and the basilisk parts are a valuable resource that can help Snape in his own quests, like for potions or in dealing with certain dark magic threats.
Another possibility is that the main antagonist could be some sort of ancient evil that is awakened by the events in the fanfic. Dumbledore might have given Snape to Harry because he sensed this new threat. This ancient evil could have powers that are unknown to the modern wizarding world and it would be up to Harry and Snape to figure out how to defeat it.
Perhaps he simply lost his creative inspiration. Fanfiction writing often requires a constant flow of new ideas, and if he's been through a lot of stress or has had some personal issues, like his complex relationship with Lily still haunting him, it could dry up his well of creativity.
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.
Perhaps Harry has a hunch that Snape is involved in some secret project or fight against the dark forces that requires basilisk parts. Harry, being the hero he is, decides to provide Snape with the parts to aid in this unspoken cause. It could also be a sign of Harry's growing maturity and his ability to see beyond their past differences.
One possible plot could be that Snape discovers he has a unique magical ability that allows him to carry a child. He might keep it a secret from everyone in Hogwarts due to fear of ridicule. As he nears the time of giving birth, he would have to find a secluded place within the castle or in the Forbidden Forest. He would endure the pain alone, using his own potions and magic to ease the process as much as possible.
The main characters are obviously Harry and Teddy. Harry Potter, the famous wizard who defeated Voldemort, and Teddy Lupin, the son of Remus Lupin and Nymphadora Tonks. Harry is likely in a mentoring role here, while Teddy is the recipient of whatever knowledge or advice Harry is giving.
It could signify a turning point in their relationship. They've had a rocky relationship in the original series, so this could be a moment of reconciliation. Snape might finally be seeing Harry as more than just James Potter's son.