Praise be to Allah.
We addressed this question to Shaykh ‘Abd-Allaah ibn Jibreen, who wrote back to us with the following response:
The correct view is that there is no sin on the child of zinaa because of the sin of his parents, because he did not do anything wrong. The (burden of) sin is on his parents. On this basis, he can call himself after his father who has acknowledged his paternity, so that he can prove that he belongs to him and his state (in official papers). Or he can call himself after his mother who gave birth to him, because this is his line of descent; then he will belong to her tribe and state. He should also do righteous deeds and be upright in his conduct and commitment to Islam. What his parents did should not cause him any harm. A person who is hindered by his bad deeds will not be helped by his lineage [i.e., a person’s origins do not count for anything if his actions are bad]. And Allaah knows best.