Perhaps there was also a sense of self - preservation. If Lucius were to harm Harry's daughter, it could lead to a major upheaval in the wizarding world. Snape, being in a position where he had to maintain some sort of balance, might have stepped in to prevent chaos that could ultimately endanger his own life or his standing in the wizarding community.
Snape could also be acting out of a sense of justice. Lucius was a Death Eater, and Snape, despite his own past as a Death Eater, had turned against Voldemort. Saving an innocent, especially Harry's daughter, from Lucius would be in line with his new allegiance to the side of good.
Snape might use his skills in potions or dark magic to surprise Lucius. For example, he could quickly brew a potion that temporarily immobilizes Lucius, allowing him to snatch Harry's daughter away to safety.
One possible motive could be that Snape is still deeply influenced by his loyalty to Voldemort in this fanfiction. Maybe Voldemort has some hold over him that forces him to do the unthinkable. Another reason could be a misunderstanding. Snape might believe that Harry has turned to the dark side or is about to, and he decides to take preemptive action to stop what he thinks will be a great evil.
One possible motive could be a misunderstanding. Maybe Snape is framed for something really bad, like a dark magic plot against Hogwarts. Harry, in his haste to protect the school and his friends, believes Snape is guilty and takes drastic action. Another reason could be if Snape accidentally endangers someone Harry loves dearly, like Ginny or Ron, and Harry's emotions get the better of him in the heat of the moment.
Perhaps in the fanfic, Snape discovers a powerful and dangerous magic that he wants to use for what he believes is the greater good, but Harry sees it as a threat. Harry has always been wary of Snape, and this new discovery could be the tipping point. Also, if Snape were to make a deal with a dark force in order to gain knowledge to defeat Voldemort but Harry misinterprets his intentions, it could lead to Harry killing Snape.
One possible motive could be Snape's complex relationship with Harry's father. Snape might see Harry as a way to get back at James in some twisted way. He could also be acting out of a sense of duty to the Dark Lord, believing that taking Harry by force is part of some larger plan to gain power or favor.
Another motive could be related to his complex relationship with Dumbledore. Dumbledore might have asked Snape to look out for Draco, and Snape, being obedient to Dumbledore in many ways, would do so to fulfill his request. Also, Snape might believe that protecting Draco is part of a bigger plan to maintain the balance in the wizarding world.
In fanfiction, Snape saving Harry often showcases a complex side of Snape's character. It might be a redemption arc for him, as in the original series, Snape had a very ambiguous relationship with Harry. In these fanfictions, it could be due to his love for Lily that he finally decides to fully protect Harry, which is a very interesting take on his character.
One possible plotline is that they are all framed for a crime they didn't commit. To clear their names, they must team up. Harry's quick thinking, Snape's ability to deceive when needed, and Lucius's connections all play a part. As they work together to find the real culprit, they form a bond of mutual trust and respect, which is the foundation of their new - found mate - ship.
One possible motive could be his love for Harry's mother, Lily. Even in the fanfiction, this could be a driving force. He might see helping Harry as a way to atone for his past actions towards Harry.
Perhaps the fanfiction focuses on a parallel universe where Snape, Harry and Lucius are all on the same side from the start. They are part of a secret group that is trying to reform the wizarding world's education system. Their different perspectives - Snape's strictness, Harry's compassion and Lucius' cunning - combine to create a powerful force for change.