Perhaps in the fanfiction, the Basilisk has a hidden power that can only be unlocked by Harry's wand. A prophecy states that Harry must use his wand to unlock this power and use it for good. However, there are those who want to use the Basilisk's power for evil. Harry has to race against time to figure out how to use his wand to unlock the power and keep it out of the wrong hands. This would involve a lot of exploration of the Hogwarts secrets and battles against dark forces.
A fanfiction plot could revolve around Harry being cursed by the Basilisk's stare, but his wand has a special property that can slowly reverse the curse. As he tries to heal himself, he discovers that the wand has a deeper connection to the Basilisk than he thought. The wand was once used to create the Basilisk, and now it holds the key to its destruction. Harry must learn about the wand's history and use that knowledge to defeat the Basilisk once and for all.