Well, a frequent theme is the growth of their relationship against the backdrop of the wizarding war. They might start dating secretly during dangerous times and have to keep it from their enemies. Also, the theme of self - discovery within the relationship is common. As they date, they learn more about themselves, like Harry learning to open up more emotionally and Hermione learning to relax a bit. Moreover, there are often themes related to how their relationship affects the group dynamics of the Golden Trio and other friends.
One common theme is the exploration of their intellectual connection. Since both are very intelligent, fanfics often show how this aspect enhances their relationship. Another is dealing with jealousy from other characters, especially Ron. And sometimes, there are themes about them using their combined knowledge to solve big problems in the wizarding world.