Praise be to Allaah.
It is haraam for a person to call himself after anyone
other than his father, or after people to whom he does not belong. This was stated in the
saheeh hadeeth narrated from Abu Dharr, who heard the Messenger of Allaah
(peace and
blessings of Allaah be upon him) say: There is no one who knowingly calls himself
after someone other than his father, but he is guilty of Kufr
(Narrated by al-Bukhaari, 3371; Muslim, 61).
According to the version narrated by al-Bukhaari: who calls
himself after people to whom he does not belong
Falsely attributing a person to people he does not belong to results in
violation of the laws of Mahram (close blood ties dictating who is and is not permissible
for marriage), inheritance, custody, guardianship in marriage, etc.
With regard to what should be done, we put this question to Shaykh
Muhammad ibn Saalih al-Uthaymeen, may Allaah preserve him, who answered as follows:
She cannot take the name of anyone other than her family, and it is not
permissible to do so. She has to change her name.
If she does not know her fathers name or family, she can call
herself by a general name such as Faatimah bint Abd-Allaah or Faatimah bint
Abd al-Rahmaan.
And Allaah knows best.