One common trope is the outcast status. Since Hera is not the most liked among the gods in the Percy Jackson universe, her son is often made an outcast among the demigods. Another is the special powers. He might have some unique abilities related to Hera's domain, like control over family bonds or something similar.
A lot of these fanfics play with the trope of a hidden heritage. The son of Hera might not know about his true parentage at first and then discovers it in a dramatic way. Also, there's the trope of having to prove himself. Because of his connection to Hera, he has to constantly show that he's not like his mother in the negative ways people perceive her.
In many of these fanfictions, there's a trope of an uneasy alliance. The son of Hera has to form alliances with the other demigods who are initially skeptical of him. For example, in some stories, he has to work with the children of Zeus, which is quite difficult considering the enmity between Zeus and Hera. This creates a lot of tension and interesting plotlines.