Is it permissible to remain with a husband who is said to be the son of Zina, and what are the implications of that?

Interfering with lineages is dangerous, and spreading rumors and claims about them contradicts Islamic law. The basic principle is that a child belongs to the husband whose wife gives birth to him, as the Prophet, peace and blessings be upon him, said: "The child is for the bed (i.e., for the husband of the woman who bore him)." Paternity is not denied except through mutual imprecation (li'an). You must disregard what your relative says and advise her, warning her against slandering Muslims and accusing them of indecency without proof. This is a grave sin, an act of ignorance (Jahiliyyah), and a false accusation that necessitates a prescribed punishment (hadd), as stated in His Almighty's saying: "And those who accuse chaste women and then do not produce four witnesses - lash them with eighty lashes."