How can my cousin expiate her sin for supplicating against her oppressive father, especially since she abhors injustice and strives to be righteous?

Advise your cousin as follows: She should supplicate frequently and exert efforts that aid in steadfastness, such as seeking beneficial knowledge, performing righteous deeds, attending gatherings of goodness, and keeping company with righteous women. This is based on Allah's saying: "And those who strive for Us - We will surely guide them to Our ways. And indeed, Allah is with the doers of good" (Al-Ankabut: 69).

Repentance is obligatory for all sins by abandoning them, regretting them, and resolving not to return to them. If the sin involves a right of another person, she should seek their forgiveness.

As for her father: It is not permissible to abuse him or supplicate against him, no matter how much he wrongs her. Such an act is an act of disobedience to parents, from which one must repent. She should supplicate for him for goodness and righteousness and strive for reconciliation, involving people of good standing to mediate.