In this kind of fanfiction, Percy might challenge Zeus over issues like the unfair treatment of demigods. For example, Zeus could be overly harsh on demigods who make minor mistakes while the gods themselves get away with much worse. Percy, being a hero, would stand up for his fellow demigods.
One common theme is the exploration of forbidden love. Since the world of Percy Jackson has its own set of rules and norms, a lesbian relationship between characters might be seen as going against the traditional ideas in that world. Another theme could be self - discovery, where a character realizes her true feelings for another female character and has to come to terms with her own identity.
One idea could be exploring Percy's life if he had a different godly parent. For example, if he was a son of Hades, how would his powers and his relationship with the other demigods change? Another idea is to write about a time - travel adventure for Percy and his friends, where they go back to the era of the first Olympians.
One possible plot could be that the entire Greek pantheon is thrown into chaos. Percy might be seen as a huge threat, and other gods and goddesses would either try to hunt him down or form alliances with him out of fear or a desire for power. Demigods would be forced to take sides, leading to massive battles among them.
Annabeth is also a very popular choice. Her intelligence and her relationship with Percy are often explored in fanfics. Writers might create scenarios where she uses her smarts to solve complex problems or where her relationship with Percy is put to the test in new and interesting ways.
One idea could be Percy having to face a new and powerful monster that is a combination of different mythological creatures. For example, a chimera with the head of a hydra. This would create a unique and challenging battle for him.
A possible plot could be Percy discovers Zeus has been secretly plotting against some of the more powerful demigods. Percy then challenges Zeus to a duel of sorts, with the fate of the demigods at stake. If Percy wins, Zeus has to stop his plotting and give the demigods more freedom.
There could also be other characters like Annabeth. She is Percy's friend and often provides support and wisdom. In the context of this fanfiction, she might be involved in helping Percy figure out the mystery of the Zeus lemon. Maybe she uses her intelligence to research ancient texts related to the significance of the lemon in relation to Zeus.
Yes, there are quite a few interesting ones. You can start by looking on fanfiction platforms like Archive of Our Own. Some fanfics might explore Percy's relationship with his divine parents in unique ways, or perhaps put him in new and exciting adventures that are different from the original series.