Although it makes sense to change it to "what my name is," because your name is always going to be the same, it all has to be in the past tense.
"The teacher asked me what my name was."
It is "he asked where she worked now," because he asked her in the past, but where she works is current. Then would mean he asked her where she worked in some time in the past, either at the time the conversation went on or a time before that.
[Not a teacher]
Student or Learner