What is the ruling on a son continuing to advise his mother about what he deems correct, if that makes her angry and causes her to accuse him of being biased towards his father?

Part of perfecting filial piety towards one's mother is for her child to explain the truth to her regarding the resolution of family problems, adhering to the etiquette of advice without lying or flattery. This is because the truth is more deserving of being followed, even if it is bitter, in accordance with the saying of the Prophet, peace and blessings be upon him: "Whoever seeks Allah's pleasure through the people's displeasure, Allah will be pleased with him and make the people pleased with him. And whoever seeks the people's pleasure through Allah's displeasure, Allah will be displeased with him and make the people displeased with him."

As for continuing to advise her after she has shown anger, scholars have prohibited this. The Sunnah concerning enjoining good upon parents is to do so once. If they accept, then all the better. If they dislike it, one should remain silent and occupy oneself with supplication and seeking forgiveness for them.