Praise be to Allah.
It is not permissible for a woman to take her husband’s name or his family name because that is attributing oneself to someone other than one’s father, and imitating the kuffaar from whom this custom was adopted.
Al-Bukhaari (3508) and Muslim (61) narrated from Abu Dharr (may Allaah be pleased with him) that he heard the Prophet (peace and blessings of Allaah be upon him) say: “Any man who knowingly attributes himself to someone other than his father is guilty of kufr. Whoever claims to belong to a people when he has nothing to do with them, let him take his place in Hell.”
“when he has nothing to do with them” means, when he has no lineage among them, as is highlighted in some reports.
Based on that, the husband has no right to force his wife to do that, and if he forces her to do it she should not obey him, because it is obedience to a created being which involves disobedience to the Creator. So she should persist in her refusal and explain to him that it is haraam, and look for Islamically acceptable means of establishing her rights from a legal point of view.
For more information please see the answer to question no. 6241 and 1942.
And Allaah knows best.