One common theme is loyalty. Hayate is always loyal to Nagi in the series, and this is often explored further in fanfiction. For example, how far he will go to protect her.
One interesting fanfiction could be where Hayate gets involved in a special mission within the Naruto universe that was not in the original story. It might explore his combat skills in a new and exciting way.
One great 'Nagi x Hayate' fanfiction is 'A Chance Encounter'. It beautifully portrays the relationship between Nagi and Hayate, with vivid descriptions of their interactions. The author does an excellent job of showing Nagi's dependence on Hayate in a really heartwarming way.
Another popular trope is the 'found family' one. Hayate could form bonds with other characters in the Naruto world, creating a sense of family. This can be really heart - warming in the context of the often - dangerous Naruto universe.