One key element is the challenges Leto faced when she was pregnant. Zeus had impregnated her, but she was persecuted by Hera. She had to wander in search of a place to give birth safely.
One possible plot could be Percy's journey to discover his true place among the demigods. Since he is the son of Leto and Zeus, he might face unique challenges. For example, he could be caught in the middle of power struggles between different camps of demigods. He may also have to deal with the expectations and rivalries that come with having such powerful parents. His powers, which could be a combination of those associated with Leto and Zeus, might make him a target or a valuable ally.
His powers might include the ability to control the weather like Zeus. So he could summon thunderstorms or clear skies at will. From Leto, he could inherit a connection to the moon. This might give him enhanced senses during the night or the ability to draw power from moonlight. He could also have some form of protection or blessing related to his mother's association with motherhood and fertility in mythology.
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.
When writing Percy, think about his backstory. He grew up knowing his divine heritage, which could have both positive and negative impacts. He might have been bullied by other demigods out of jealousy, so he has a bit of a guarded side. But he also has a strong sense of justice, inherited from Zeus, which makes him always ready to stand up for the weak. His relationship with his parents should also be explored. He could have a closer bond with Leto as she was more present in his life, while Zeus was often distant due to his many responsibilities in Olympus.
Common themes include identity crisis. Percy has to figure out who he is now that his parentage has changed. There are also themes related to the Olympian hierarchy. With Zeus as his father, he might be thrust into a different position in the power structure. Additionally, themes of loyalty can come up. Who does he stay loyal to - his old friends and family or his new divine family? These themes make the fanfictions engaging and full of potential for complex storylines.
I'm not sure specifically what that particular fanfic is about as I haven't read it. It could potentially be a fan - created story that focuses on the relationship between Zeus and Hera with some sort of unique or interesting twists. Maybe it reimagines their mythological relationship in a modern or different setting.
Another idea is a story where Percy discovers a hidden prophecy related to his birth. In this fanfiction, he might have to team up with some unlikely allies, like a group of minor gods who are also interested in the outcome of the prophecy. The story could involve Percy learning more about his parents' relationship and how it has shaped his own destiny.
One possible plotline could be that Annabeth discovers some sort of secret that Zeus is hiding, and a lemon (which could be a magical lemon or just a symbol) plays a role in uncovering the truth. For example, the lemon might be a key ingredient in a potion that reveals hidden memories.