The lady is Florian’s mother, Marguerite. Her name may never actually be brought up in the story though. XP

Will probably be making next week my break time.

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↓ Transcript
Suddenly a door opens nearby and two people walk out - a crying old lady in black attire named Marguerite, the young man's mother, and a somber priest. The young man immediately shuts the violin case closed upon seeing them.

The priest is holding what appears to be a cross and a receptacle for oil while Marguerite cries into a handkerchief beside him. He smiles sadly at the young man.
Priest: <<Jean-Antoine Doré has been given his Last Rites.>>*
*Brackets mean translated from French

The young man stares at him vacantly.
Priest (offscreen): <<Now he wishes to see his son, and to hear one final song from him.>>

We see the young man is now walking towards the open door at the end of the hallway, violin case in hand, as the priest and Marguerite stand aside.

However, Marguerite grabs the young man's arm, stopping him.
Marguerite: Florian...