1.She said her name is Anne (as we know to make one sentence only need one or more subject but one principle verb,in this sentence said is not main verb, SAID is here used as verb-3.Also we know that before verb-3 if no auxiliary verb are used that verb remain inactive verb.So is the main verb. Also you can think other way,our name never change as what kept when we birth.So, if we use was that means in past time her name was Anne,now changed to another name. I THINK YOU UNDERSTAND.