They could also be used in some form of dark magic defense. Snape might use the basilisk parts to create spells or enchantments that can protect Hogwarts from dark creatures or dark wizards. Maybe he could create a magical barrier using the basilisk parts' power that repels any dark magic approaching the school.
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.
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.
He might be initially suspicious. Snape is a cautious man, and he may think that Harry has some ulterior motive, like setting him up. But after realizing Harry's sincerity, he could be grudgingly grateful.
They can be a source of mystery. Writers might use the basilisk eyes to hide secrets. For instance, there could be a hidden message that can only be seen when looking into the basilisk eyes under a certain magical condition in a fanfic.
In some Harry Potter fanfiction, the basilisk auction at Gringotts could be a very dangerous and illegal affair. It might involve dark wizards trying to obtain the basilisk for its powerful and deadly abilities, like its petrifying gaze. Maybe it's a way for the author to explore the seedy underworld of the wizarding world where such dangerous magical creatures are traded on the black market.
In some fanfics, the basilisk can be a source of great power for Harry. He might absorb some of its magic or gain a special ability after a close encounter with it.
In some fanfictions, basilisk eyes can be a source of dark magic. They might be used by dark wizards to cast powerful and malevolent spells. For example, a dark wizard could use the basilisk eyes to create a fear - inducing spell that spreads across Hogwarts.
Maybe he needs money for some special purpose, like funding the DA activities more privately without relying on donations. He could also be thinking about using the money to research more about dark creatures in order to better protect Hogwarts in the future.
One challenge could be the basilisk's own will. Even if Harry has the ability to control it, the basilisk is a powerful and ancient creature with its own instincts. It might resist Harry's control from time to time, especially when it comes to situations that go against its natural behavior, like being made to not attack innocent students.
It could play a major role as a plot device. For example, it might be a key ingredient in a potion that can cure a deadly disease in the wizarding world.