Given Name
Riza Hawkeye is a woman structured on independence. She is more than capable of leading men through a battle, no matter if it was through a sandy combat zone or through her Colonel's paperwork. She equates dependency for others but requires none for herself.
Her given name is testament to her individual strength, a name no one dares to sully or take for granted. It symbolizes Riza as a strong soldier, a sharp shooter, but seldom as the gentle woman she is.
She is rarely called by her first name, so it is generally assumed that Hawkeye (First Lieutenant, Lieutenant, ma'am) is all she answers to. She introduces herself and in turn is introduced with her surname, something no one takes any notice of.
When Roy Mustang proposed to her their engagement had been kept secret until pretenses could absolutely be hidden no more, but many still believed that she would keep her last name as she's lived decades with it. First Lieutenant Mustang surprised everyone when she discreetly signed a few forms to officially changed her last name.
Because, you see, the name you're born with is intrinsic, a name you have no choice in accepting. It links your past with your present, and carries off into the distant future. But when someone offers you their name they're giving you a choice to start a new family, one that takes in both your given and offered name. One is given when your parents lovingly christen you with the family name to uphold, and the other is offered with love and permission of acceptance into the rest of your life.
And Riza's personally liked horses better than birds, in any case.