It could be Aphrodite. Given her domain of love, it would be an interesting take. Percy might be charmed by her beauty and grace, and they could end up having a child. The child might inherit Aphrodite's charm and Percy's courage.
Maybe the plot would revolve around the relationship between Percy and the goddess. It could show how they met, fell in love, and decided to have a child. Then it might explore how the other gods and goddesses react to this union and the birth of the child.
It could be Annabeth, who was made a goddess in some interpretations. Their kids would be really smart and brave, inheriting Percy's heroism and Annabeth's wisdom.
Once upon a time, Percy, the god, watched over his child from afar. The child, named Alex, was growing up in a small village. Percy could only visit in secret, for the laws of the gods forbade direct interference. One day, when the village was in danger from a great flood, Percy had to decide whether to break the rules to save his child and the villagers. In the end, he sent a sign to Alex, guiding him to build an ark and save everyone.
One idea could be that Percy, as a god, has a child with a mortal. The child might inherit some of Percy's powers but also face the challenges of being a demigod in a modern world. Maybe the story could focus on how Percy tries to protect his child from the various threats that come with being part of the godly world while also teaching the child to embrace their powers.
It could be Aphrodite. Since she is the goddess of love, it would be an interesting twist. Percy might initially resist her charm but eventually be won over by her true self. Their marriage could bring a new understanding of love to the world of the fanfiction.
It could be an old acquaintance that Percy had a brief relationship with. Maybe someone he met during one of his adventures and then lost touch with, but she later discovers she's pregnant and decides to keep the child a secret from Percy for various reasons.
One possibility could be that Percy has been leading a double life. Maybe he had a relationship in the past that he kept hidden, and now the existence of the child is coming to light. The story could explore how he copes with this new reality and how it affects his relationships with his friends and family.
The child could have the ability to control water like Percy, but in a more extreme way. Since Percy is now a god, the child might be able to create entire oceans or control the water cycle on a global scale. It could also have enhanced strength and speed, typical of demigods, but on a much higher level.