Star Morrison pleads the fifth when asked to state her name for the court. Then we learn that she's undercover FBI agent Dana Lewis... so if she had given the name of her assumed identity she'd have perjured herself and possibly caused a mistrial.
This is pretty clearly stated as her reasoning in the scene with Munch. It's not clear whether it would have caused a mistrial or not, but it could've gotten Dana herself in a lot of trouble. It also gives her a convenient way out of testifying, because her testimony is a much bigger potential issue: if she tells the truth, she could compromise her operation by losing the trust of her targets, but if she commits perjury within her actual testimony, that's much worse than just lying about her name, not to mention she wouldn't want to be responsible for a child killer walking free. By doing what she does, she can keep the trust of the group because she can make them think she's just being difficult for the sake of obstructing the process, while not actually having to lie or give testimony that could get Ackerman off.
