Praise be to Allah.
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.