In the fanfiction's plot, it's possible that some strange events have been occurring in Hawaii that are linked to Percy. Artemis, being vigilant about maintaining order in the demigod realm, feels compelled to find Percy. These events could be related to the local Hawaiian mythology that has somehow intertwined with the Greek demigod world in the fanfic. Maybe there are new monsters emerging that seem to be connected to Percy, or there are disruptions in the natural balance of the Hawaiian islands that Artemis believes Percy can help explain or fix.
In such a fanfiction scenario, perhaps Artemis could initially be suspicious of Percy's presence in Hawaii. She might think he's there for some hidden agenda related to the demigod world. Maybe she'd keep an eye on him secretly at first.
Artemis could be on a quest to stop an evil force that she believes Percy might be involved with, whether willingly or not. The research lab could be a place where this evil force is planning something big, and Percy's presence there might be a key factor. So, she enters the research lab to find Percy and figure out his role in the whole situation.
Well, she could betray Percy by revealing his secrets to his enemies. Maybe she's forced to do so under some kind of magical compulsion or threat.
Percy could lose trust in the gods completely. After Artemis' betrayal, he might feel that he can't rely on any of them anymore.
Maybe Artemis is on a hunt when she stumbles upon Percy. She might be initially cautious, as she's a goddess and Percy is a demigod. She could question him about his presence in Texas.
Percy could also react with a sense of determination. Instead of being defeated by the injury, he sees it as a challenge. He starts to figure out what made Artemis hurt him in the first place. If it was a misunderstanding, he tries to clear it up. If it was something more malicious, he starts to plot his revenge. He uses his own wits and the powers he has to either heal himself or find a way to make Artemis see the error of her ways. He might also seek the help of his friends, like Annabeth or Grover, to come up with a plan to deal with the situation.
The Hunters of Artemis are very likely to show up. They are Artemis's loyal followers and would surely be involved in the search for their mistress. Annabeth could also be there. She's Percy's friend and has great wisdom, which could be useful in devising a plan to save Artemis. Grover might be another character. His ability to sense things could help in tracking Artemis or finding clues about her whereabouts.
Perhaps a story where Percy is chosen by the Fates to mediate between Hestia and Artemis. There has been some sort of misunderstanding between the two goddesses. Percy, with his charm and unique perspective, tries to get them to see eye to eye. He arranges meetings in different places - sometimes in Hestia's warm and cozy domain, and other times in Artemis's wild and untamed forests. Through Percy's efforts, they not only resolve their differences but also realize that they can work together for the greater good.
One reason could be the popularity of Percy as a character. Fans might be more inclined to write about him in a dominant role because they like him better. So, they create fanfictions where he dominates Artemis.
In some fanfictions, Percy could be made Artemis' son for creative and often unique plot twists. Fanfiction allows for a lot of creative liberties, so it's not unheard of. However, in the original Percy Jackson lore, there's no such relation.