What can I do to seek forgiveness for my mother, who gave birth to my sister out of wedlock after my father's death, knowing that she was repentant, pious, and virtuous?

The mother and daughter must be concealed, as stated in the Hadith: "Whoever conceals a Muslim, Allah will conceal him on the Day of Resurrection." If a woman is married to a man and gives birth to a child within the possible gestation period, the child is attributed to the husband. No one is permitted to deny his lineage unless the husband disavows him and performs li'an (oath of condemnation). Even if both parents agree to deny the lineage, it will not be negated except through li'an.

As for what can be done for the mother so that Allah may forgive her, the most important actions are supplication (du'a), seeking forgiveness (istighfar), giving charity on her behalf, and performing Hajj and 'Umrah if possible. It is also permissible to give charity on her behalf and fulfill her vows after her death. It is recommended to marry off the daughter when she reaches maturity to a man who will keep her away from contact with her paternal uncles, as a precautionary measure.