No, she needs to keep her maiden name. There is nothing in the Islamic traditions to indicate that a woman should take her husband’s name, rather this is an innovative matter that is not approved of by Islamic law. The wives of the Prophet (peace and blessings of Allah be upon him) are the Mothers of the Believers, and the Prophet (peace and blessings of Allah be upon him), who was the noblest of people, married them but not one of them took his name, rather each of them kept her father’s name even if her father was a non-muslim. Similarly, the wives of the companions and those who came after them did not change their names.
There is no blood tie between the husband and wife so she cannot take his surname as if she is part of his same lineage. She may get divorced, or her husband may die, and she may marry another man. Which husband’s last name will she keep, or will she keep changing her last name with every new marriage? The answer is of course not, and that is why Islam wanted each one of us to keep his or her surname as is to preserve and protect the lineage of people.
Allah says (interpretation of the meaning): “Call them (adopted sons) by the names of their fathers” [al-Ahzaab 33:5].
Narrated from Abu Dharr (may Allah be pleased with him) that he heard the Prophet (peace and blessings of Allah 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.” Al-Bukhaari (3508) and Muslim (61)
The Prophet (peace and blessings of Allah be upon him) said: “Whoever calls himself by other than his father’s name (or attributes himself to someone other than his father), will be cursed by Allah, the angels, and all the people.” (Reported by Ibn Maajah, 2599)