The relationships could develop in a very dynamic way. Let's say Percy has an encounter with a powerful enemy that threatens both Athena's temples and Artemis's forests. Percy decides to fight this enemy, and in the process, he calls upon both goddesses for help. Athena provides him with knowledge and strategic advice, while Artemis sends some of her hunters to assist. After they defeat the enemy together, Athena starts to view Percy as more than just a reckless demigod. She sees his potential for leadership. Artemis, on the other hand, develops a bond with Percy because of his courage and respect for her hunters. This event could be the starting point for a deeper and more complex relationship between the three in the fanfiction.
Percy could interact with them in a variety of ways. For example, he might approach Athena in her temple seeking knowledge about the lemon. Athena, being the goddess of wisdom, could provide him with some cryptic clues. With Artemis, Percy could be on a joint mission where the lemon is somehow involved. Artemis, with her fierce independence, might initially be hesitant to work with Percy but then realizes they need each other.
One idea could be Percy having to go on a quest that involves both Athena and Artemis. Maybe he has to retrieve an ancient artifact that is sacred to both goddesses. Another could be a story where Percy gets caught in the middle of a feud between Athena and Artemis over a new demigod's loyalty. For example, the new demigod shows great skills in both wisdom (Athena's domain) and hunting (Artemis's domain), and the two goddesses start to compete for the demigod's allegiance, with Percy trying to keep the peace.
One plot idea could be that Percy finds himself in a situation where he has to seek the wisdom of Athena and the protection of Artemis while being mysteriously linked to a lemon. Maybe the lemon is a magical object that only he can unlock its power, and Athena helps him figure out how to do it while Artemis guards him from those who would steal it.
In these fanfictions, Percy often retains his bravery and loyalty. His more impulsive nature can create interesting dynamics with Athena's calculated wisdom. For example, he might rush into a situation while Athena is thinking of a strategic plan.
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.
One great Percy Jackson and Athena fanfiction is 'Athena's Champion'. It really delves into the relationship between Percy and Athena in a unique way. The author portrays Athena as having a more hands - on role in Percy's adventures, guiding him with her wisdom in ways that are not explored in the original series.
One idea could be a story where Percy teams up with the Athena Hunters on a quest. Maybe they are searching for a lost artifact that has the power to change the balance between the gods. Percy's water powers could complement the skills and knowledge of the Athena Hunters.
Another plot could involve Percy getting a vision from Athena about a future threat to the demigod world. But the vision is so cryptic that he has to work closely with Athena to decipher it. Along the way, they build an unexpected bond, different from his usual relationships with other gods. Maybe Percy starts to see Athena not just as a powerful deity but as a mentor figure, and Athena starts to respect Percy's tenacity more than she expected.
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.