Perhaps it's an old enemy that has come back with a new plan. For instance, a Titan or a powerful monster that Percy thought he had defeated before. They might have found a loophole in the rules of the demigod world and managed to banish Percy to Alaska as a form of revenge and to keep him out of the way while they carry out their other evil plans.
It could be a new enemy. Maybe a powerful and vengeful demigod who is jealous of Percy's fame and power. This demigod could have found a way to use some ancient magic or some political maneuvering within the demigod world to get Percy banished to Alaska.
Well, he might face a lot of challenges. Alaska is a cold and harsh place. He could encounter new mythical creatures that are native to that region in the fanfiction. Maybe he has to find a way to survive the extreme cold while also dealing with any enemies that banished him there in the first place.
It could also be an ancient prophecy that is misinterpreted. The Oracle might give a prophecy that the gods misunderstand as Percy being a threat. So, in an effort to prevent the prophecy from coming true in what they think is the wrong way, they disown him.
It could be an old enemy like Luke. He has always had a vendetta against Percy and would not hesitate to cause him harm. Maybe Luke has some new evil plan and hurting Percy is part of it.
He might meet some native Alaskan spirits. These spirits could be like the nature spirits he has encountered before but with their own unique Alaskan characteristics.
In an Alaska - based Percy Jackson fanfiction, Percy could be on a quest that leads him to the frozen tundra. He might discover an ancient Greek artifact that has been hidden there for centuries. This artifact could hold the key to preventing a major disaster, like the awakening of an old and powerful enemy. Along the way, he could team up with some local Inuit demigods who have their own unique powers and knowledge of the Alaskan wilderness.
He might rely on his powers. Percy is a son of Poseidon, so his control over water could be crucial. He could use water sources in Tartarus, if any, to heal himself, create barriers, or even find a way to escape. For example, he could manipulate the Styx water to his advantage.
In some fanfictions, Percy might be banished because he goes against the traditional ways of the demigod society too much. For example, if he tries to form an alliance with a group that the other demigods consider enemies. His actions could be seen as a threat to the stability of their world, so they banish him to keep the peace.
It could be Ares. He is often portrayed as a violent and unpredictable god. Percy might banish him if he goes too far in causing wars and chaos among mortals. His banishment would have a huge impact on the battlefields of the mortal world.
Obviously, Percy Jackson himself. And Annabeth could also be there, helping Percy understand the new environment. There might be some new characters like an Alaskan native who is a seer and can guide Percy in his adventures.
Obviously, Percy Jackson himself. Then there could be Annabeth, his girlfriend, who might come to Alaska to support him or offer her strategic advice. Grover could also show up, using his nature - sensing abilities in the Alaskan wilderness. There might be new characters too, like local Alaskan demigods or spirits that Percy has to interact with, for example, a spirit of the northern lights who has its own powers and agenda.