Around the world, it is a norm for a woman to change her surname once she is married. She becomes Mrs. someone. It may seem sweet to some to belong to a man. However, Islamic jurisdiction Sharia opposes this. All the Mothers of Believers, the wives of the Prophet were known by their father's name. It was an honour to be Prophet's wife but why didn't they change their surnames? Simple, you do not change your surname after marriage according to Islamic traditions.
Furthermore, the Quran points out that adopted children should be called after their father's name likewise is for a married woman.
"Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah..." [al-Ahzaab 33:5].
In addition, Hadith states that every human being will be raised on Day of Judgement by his / her father's name.
"On the Day of Resurrection, each betrayer will have a banner raised beside him, and it will be said, this is the betrayer of So and so the son of So and so." (Narrated by al-Bukhaari, 5709, and Muslim, 3265).
When I ponder on this, it does make sense. What if the woman marries twice? Will she keep changing her name? What is her true identity then? Today she is Mrs Ahmed tomorrow she is Mrs Muhammad? Keeping your father's name will maintain her identity irrespective of her marriage(s).
I know for a fact in many Arab countries women do not change their surnames after marriage. However, in countries such as Pakistan and India, a woman does change her surname. I, for one didn't. I am still holding my maiden surname. At first, my husband was very angry with me and repeatedly questioned me why I didn't want to change my name. Even, my father in law thought I was not committed to this marriage and I would eventually ask for a divorce. But I stayed firm in my belief. Now years after, my husband and father in law have no objections and actually understand my viewpoint.
Take a stand and follow the Sunnah. The world may turn against you but when you are following the Straight Path all goes well at the end.