Most Somali names come from one of two sources: The Qur'an (such as Mohamed, Khadija, and Fadumo) or Somali tradition (such as Beydan, Burale). I have seen a few nontraditional names, and my guess is there will be more of these in the children and the grandchildren of the Somali Diaspora.
Somali women traditionally do not change their names when they get married.
Somalis do not have family surnames in the European sense of the word -- that is, a single family name that follows the male line(s) of the family down through generations. Instead, there is a system of naming that is more of a chain.
There are usually three names, and with each generation, the last of the three drops off and a new "first name" is given.
The best way to explain is through an example.
Let's take the name Abdi Ali Hassan.
Abdi is the individual's personal name -- his "first name", to use U.S. terminology.
Ali is his father's "first name" and Hassan is his grandfather's "first name".
If Abdi marries Fadumo Issak Hersi, her name will be unchanged. (Note: the children they have together will be considered to belong to Abdi's clan and tribe.)
In time, Abdi and Fadumo have children -- let's say a boy and a girl.
Traditionally, the first male child is named Mohamed, and this is still often the case. We'll call the girl Hawo.
So Mohamed and Hawo are the "first names".
Then the father's first name -- Abdi.
Then the grandfather's first name -- Ali.
So the children are Mohamed Abdi Ali and Hawo Abdi Ali.
And when little Mohamed grows up and has children? His daughter would be, for example, Khadijo Mohamed Abdi.
And so on.
This explains why a Somali woman's children don't have the same last name as she does. Family relationships (or lack thereof) shouldn't be assumed just because people have the same last name.