One idea could be a story where a new, mysterious illness spreads in the school and the Hospital Wing is overcrowded. Harry and his friends try to find the cause while also helping Madam Pomfrey. Another could be a story about a student who has a special power that is only revealed when they are in the Hospital Wing due to an accident.
In a 'Harry Potter Hospital Wing Fanfiction', you could have a plot where a student fakes an illness to stay in the Hospital Wing to avoid a difficult exam. But then, they accidentally discover a hidden passage or secret in the Hospital Wing. Meanwhile, Madam Pomfrey suspects something is up and starts her own investigation. This could create a lot of humorous and tense moments as the student tries to keep their secret while also exploring the mystery of the Hospital Wing.
Well, perhaps a plot where a magical creature gets injured and is brought to the Hospital Wing. The students, especially those in the Care of Magical Creatures class, have to help take care of it. This could lead to some interesting interactions between the students and the creature, as well as show a different side of the Hospital Wing's capabilities. Also, a story about a forgotten spell that starts causing strange effects on students and they all end up in the Hospital Wing while the teachers try to figure out how to reverse it.