It could be a very dark and complex story. Percy might be constantly at war with himself, having grown up in a more 'good' family environment but having this dark connection to Voldemort. He could be hunted by both sides - those who fear he'll turn out like his 'grandfather' and the remnants of Voldemort's followers who might want to use him for their own ends.
There might be very few fanfictions on this exact topic as it combines two very different fictional universes. But if it exists, it could explore how the powers from the Percy Jackson world, like the ability to control water or superhuman strength, would interact with the magic of the Harry Potter world where Voldemort resides. The grandson could be on a quest to stop Voldemort from getting some powerful demigod - related artifact that could enhance his dark magic.
There could potentially be a 'Percy is the grandson of Voldemort' fanfiction. Fanfiction writers often explore very strange and creative ideas. However, without actively searching through the vast expanse of fanfiction websites, it's hard to say for certain. It might be a very niche concept where the writer is trying to subvert the normal good - vs - evil family trees in the Harry Potter universe. For example, they could be exploring how Percy's character might be different if he had such a dark family connection, or how the wizarding world would react to such a revelation.
Percy would probably rely on his friends. In the Percy Jackson series, his friends were always there for him. In this fanfiction situation, he'd need them more than ever. They could help him stay grounded and remind him of his true self, not the dark legacy of Voldemort. Also, he might learn more about his demigod side to overpower the dark side in him.
It could be a wild mash - up. Percy might inherit some of Voldemort's dark magic tendencies while still having his own demigod powers. Maybe he'd struggle with the dark side of his heritage while trying to be a hero like he is in the Percy Jackson series.
In the fanfiction, perhaps Voldemort's grandson is not as evil as his grandfather. He could be trying to redeem his family's name. Percy Jackson might initially be suspicious of him but then they team up. The grandson could teach Percy some basic magic, and Percy could teach him how to be brave in the face of real danger, not just the kind wizards face. They might go on an adventure together to retrieve a powerful artifact that could change the fate of both their worlds. This would be a great way to explore the idea of good and evil not being so black and white.
One possible plot could be that Percy Jackson's grandson discovers a portal that leads to the wizarding world. He soon learns about Voldemort's reign of terror and decides to use his demigod powers to stop him. Along the way, he teams up with some young wizards and witches who are also against Voldemort.
Percy might be a bit curious too. He's used to strange things, and meeting someone related to such a powerful dark wizard could pique his interest. He could try to figure out if the grandson has the same evil intentions as Voldemort. He'd likely keep his distance at first, but as he observes the grandson, his reaction could change.
One way could be to start with Percy discovering this strange connection. Maybe he has some strange dreams or visions that lead him to investigate. Then he could travel to the wizarding world to learn more about Voldemort. Along the way, he'd face opposition from both demigods who think he's getting too involved with dark magic and wizards who are wary of his demigod status.
Yes, there are some. Some explore unique storylines where Harry has this unexpected relation to Voldemort. However, the quality can vary greatly. You might need to search on popular fanfiction platforms and read reviews to find the really good ones.