(Important Note: the giver is not required to take these additional preferences about setting, era, tone, etc. into account, but it might help them come up with ideas, so include as many ideas as you like)
I adore cross-gen relationships, especially professor/student. Not necessarily set while they are prof and student, if that is squicky for you (it's not for me). Maybe a student had a crush on their professor, and tries to set the flames going once they return to Hogwarts (maybe to teach themselves?) I like when the older character has some power in the relationship without being smug about it, or talking down to the younger character. Or maybe an older, more experienced professor guiding and falling for a new staff member.
Happy or sad endings are fine by me, as is dark fic or something fun. I just don't like pure fluff or crack!fic. Please keep everyone in character, no wild OOCness.
I love Wizarding history, either made up or fitting neatly into the Muggle world. Politics, women finding their way in a mens world, etc. Inventive uses of magic are great, or one partner being a Muggleborn/halfblood with the other partner being a pureblood, and both learning about how the other lives and the use of magic in their lives.
Academic lives of professors -- where do they go after Hogwarts? Do they start "apprenticeships?" Go to university?