We explain the system
Asyou may have noticed, in Hogwarts Legacy there are some strange cabinets with a circular door and a lock in the center. We have named them Daedalus locks, since they are part of a secondary mission. To open them, of course, we will first have to activate the mission.
Once we have been in Hogsmade for the first time, if we go to the Transformations courtyard, we will find a student named Nelie Oggspire. It will be her who will give us the mission. In doing so, she will explain to us that there are a series of enchanted keys called Daedalus Keys, which open these mysterious cabinets.
Hogwarts Legacy: how to open the locks of Daedalus

To open the locks of Daedalus we will have to look all over the school for some peculiar golden keys with wings. We will be able to recognize them very easily, although what indicates that they are near is a particular sound of mechanical fluttering. When we locate them, we just have to approach them.

When we do so, they will start to move, leaving a golden trail that we can follow. Once we have located and chased them (the Revelio spell shows them to us if there is one nearby), we will reach the lock.
Once in front we have to press the button indicated (in case of PlayStation or PC with Play controller) to overcome a small challenge. It will be very simple, because we just have to press the X button at the right time. And what is that? Well, just when they are in front of the lock.

By doing so we will give them a kind of punch and we will embed them directly into the lock, causing it to open and give us a badge of our house. If we collect them all, all 16, we can get a special reward through the aforementioned house chest.