Change my name with Social Security

I am recently married to my husband who is from IRQ and we will be adjusting his status. My question is that I want to legally change my name with Social Security and the DMV along with many other documents but not we are not sure which one of his last names will be considered his last name once he recieives his green card. His Surname his different from his family name...for example on his passport his given name reads Ibriham ali mohsan Sayed Mohamed and the surname reads Mohamed. It's important to both of us that we share the same last name. Thank you for your time.