I have to write about un personnage important pour moi in French. But I have no idea if I can write about my mom.
Go ahead and write about your mom.
The key here is for you. The requirement doesn't state any characteristics that would restrict the kind of person which is expected, like personnage historique, d'une série, d'un film, d'un livre, d'une pièce de théâtre, politique, mythologique or whatever so you are free to choose whoever you feel is important in your own opinion.
I have no doubt that your mother is a notable person for you.
“Une personne importante pour moi” would be your mother or any relative, but the word “personnage” is usually translated into “character” or “figure”, however, “un personnage important pour moi” is ambiguous. In the same idea, for example, if you ask a kid “who's your hero?” He could answer “my dad” or “Superman”.
So you can write about your mum as “un personnage important” but it might look a bit childish. (even if, as I said, it's ambiguous)