Her brain.
Back when she was 20, she was the kind of woman who would have two mouths available: one to screw things up and the other one to apologise. Now she's got no redeeming qualities, oh poor thing.
Viva Moreau!

Then the French should have stayed out of the Middle East and North Africa.

She "dumped" the child on the father because she thought the father would be a better parent to him than she would be. And she was right. So she did the right thing for the child.
I think she never ever wanted to have children (which I can totally understand myself), but got pregnant. She could've aborted the child (maybe not in France but somwehere else).
But she didn't.
I am sure Nicolas' childhood wasn't ideal. But whose is?

North African Muslims have been coming to France for the past several decades because there is a demand for 5b4unskilled labor there. And there is that demand for unskilled labor in large part because the native French have had a low birth rate for long before that. While Brigitte - who I love for her image and sexuality, don't get me wrong - complains about Muslims taking over the country, she has hardly done her part in keeping up the native numbers. She has had only one child, and what is more damning, she has been a very poor mother to him, famously comparing her pregnancy to a tumor and dumping the child off on his father's family so she could go back to jet-setting. Her hypocrisy on this point is quite glaring.

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Why because she doesn't want muslims taking over France and forcing their beliefs down her throat?
Good for her.