Artemis might find Percy when he's broken because of a shared emotional connection or a magical bond. Maybe she's been secretly watching over him and senses his distress. Another possibility is that a friendly deity alerts her to his condition.
Well, a common theme is probably growth. Percy has to grow from his 'broken' state, and Artemis's overprotectiveness can either aid or hinder that. There could be a theme of trust too. Percy needs to learn to trust Artemis's intentions, and Artemis has to trust that Percy can become whole again on his own terms. And of course, there's the theme of the supernatural meeting the mortal (in a sense). Percy is a demigod, but still has mortal aspects, and Artemis is a full - fledged goddess, so their interaction in this fanfiction could explore how those two worlds collide.
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.
Well, Artemis is a complex character herself. Perhaps she sees something of herself in Percy, like his determination or his sense of loyalty. Fanfiction often takes these qualities and twists them in new and interesting ways. She might be drawn to how Percy is portrayed in different fan - written stories, whether it's his heroic deeds or his more vulnerable moments. Reading Percy fanfiction could be a form of escapism for her, a way to explore a different kind of heroism outside of her own experiences.