Okay, first assumption: her House is not a Storm House.
Possible deaths: illness, old age, assassination, terrorism, accident.
It could be that Zagreus had tried to persuade her mother, who resisted and her death was 'arranged' so that her daughter would be more pliable.
(One thing that keeps coming to my mind is that the party may be able to turn some of Zagreus's servants against him if they can uncover enough of his plans.)
I think her primary character note is going to be control: self-control and the ability to control people and circumstances around her.
So how does her past instil that in her?
Obviously she felt out of control.
Options: love, death, and politics.
Ahah. Whatever happened made her House fall out of favour, and the finishing school/orphanage are her way to regain prestige.
Oh, easy. Her father ran away with a lower-class woman, making her mother go a little mad and lose control of the family's business.
Our Lady Name-to-be-Determined took over the House and, finding that its merchant business was ruined and she was nearly broke, she went to the Hooded Man.
She has two dreams: to restore her family's honour among the Houses, and to fix her mother's mind.
For the first, the Hooded Man helped her start a finishing school. It kept her House in the noble circles and gave her influence over each generation of noble women. Now families fight to have their daughters enrolled in the school, and the Lady is a Very Important Person.
....meh. I'm writing myself into a corner.
Okay, flash of idea: "Bertha" mom is one of Zagreus's test subjects. He's using his alchemical processes to turn her into a yuan-ti, which (he promises) will fix her madness.
The price of this is that our Lady also uses the chemicals on the kids in her care, but she--seeing her mother improving--can easily justify this as being good for them.
Hmm. Scratch the first bit. She came up with the finishing school and orphanage idea on her own, and went to the Hooded Man for the fix-my-mom's-brain bit.