He could meet his future self. This encounter might be full of surprises as his future self could have different views and experiences. His future self might be more jaded or more powerful.
Percy may meet the descendants of his friends. For example, he could come across a young demigod who is the grandson of Annabeth. This new character might inherit some of Annabeth's wisdom and architectural skills, but also have unique abilities of their own.
One possible plot could be that Percy discovers a magical device that sends him to the future. There, he might find a world that has been drastically changed by the actions of the gods in his time. Maybe the demigod camps have evolved into something completely different, and he has to adapt to new rules and powers.
He could meet other demigods from different regions. For example, a demigod from Scandinavia with powers related to ice.
Percy might meet an old enemy who has somehow been resurrected or changed in the future. This enemy could be more powerful and have a new plan to take over the world or destroy the demigods. Or he could meet a group of time - traveling demigods from an even further future who are trying to fix something in Percy's future that has gone wrong.
He could meet an old wizard who has been guarding the vault for centuries. This wizard could either be a friend or a foe, depending on Percy's actions.
There's a chance that Percy meets the personification of Tartarus itself. It could be a shadowy, all - powerful being that tries to break Percy's spirit. Percy, being the brave hero he is, would have to resist and find a way to either outwit or overpower this entity in order to continue his journey in Tartarus.
Perhaps Percy will meet a group of environmentalists. Given San Francisco's reputation for being environmentally conscious, they could be fighting against some sort of sea - related pollution that Percy gets involved in. He might use his powers and knowledge to assist them, and in the process, make some good friends.
Percy might run into his old friends' ancestors. This could create some really strange and exciting interactions. For instance, if he meets an ancestor of Annabeth, he could learn things about her family history that she doesn't even know.
He could meet his future self. His future self might be wiser, having learned from all the experiences in the past. They could have a deep conversation about Shinji's purpose and the mistakes he's made.
She could meet her future self. This would be an interesting encounter as they could compare their experiences and knowledge. Her future self might give her some advice on things she should do or avoid in her own time. Another person she could meet is a descendant of her friends. For example, a grandchild of Harry and Ginny who has inherited their bravery and magical skills.
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.