There could be his best friend Grover. Grover might try to find Percy after he runs away, adding an element of search and rescue to the story.
Annabeth could also be an important character. She might be worried about Percy and use her intelligence to figure out where he has gone. Maybe she even blames herself for Percy running away and is determined to bring him back.
One possibility for the 'percy runs away fanfic' is that Percy runs away to protect his loved ones. He realizes that there is a danger approaching and he thinks that by leaving, he can draw the danger away from those he cares about. So he sets off on his own, facing many challenges along the way, and in the process, he might discover new powers or abilities within himself.
One possible main character other than Harry could be a local journalist named Ben. Ben might be intrigued by Harry's story and decide to write an article about him. Also, there could be a Canadian mountie, Officer Davis, who initially is suspicious of Harry but later becomes his ally as he learns more about Harry's situation.
Another possible main character could be a rival. A person who is always competing with Percy and who might use the discovery of her secret as a weapon against her. This would add a lot of conflict to the story.
Annabeth could be a main character. She might be trying to stop Percy from wanting to die. She could use her intelligence and her deep understanding of Percy to figure out what's wrong and try to talk him out of it. For example, she could remind him of all the good things they've been through together and all the people who need him.
There could be a strict but kind music teacher. This teacher sees Percy's potential and pushes him to practice harder and improve his skills. Another character could be Percy's best friend who is always there to support him during his musical journey, even when things get tough.
There could be an old Alaskan native who becomes Percy's guide. He has a wealth of knowledge about the land and its spirits. He helps Percy understand the ways of the Alaskan wilderness.
There could be a kind old lady who takes Riley in when she first arrives in Minnesota. This old lady could become like a grandmother figure to Riley, guiding her through the new environment.
Jason is likely to be another main character. Since he has his own set of skills and knowledge, he might have an interesting relationship with Percy in this story. Maybe he's a bit jealous at first of Percy's control over the ship, but then they end up working together really well. And of course, Piper could be there too. She might use her charm - speak abilities to help the crew communicate with any strange creatures they encounter during Percy's time controlling the Argo 2.
It could be some of the more vengeful gods, like Ares. If Percy has somehow angered him in the past, Ares would be more than happy to hunt him down. Also, the gods' enforcers, such as the minor deities who do their bidding, could be antagonists too. They would be the ones doing the legwork to find Percy while the major gods give orders from Olympus.
Obviously Percy would be one of the main characters. And there could be a new friend he makes in the new place, let's say a girl named Lily who helps him explore the new town.