Instead of having her own identity, she literally took Logan's. That's not being her own person. And it's not what Wolverine would want for her, which means taking that identity is something she wouldn't have remotely wanted for herself to begin with, cause she KNOWS he wants her to establish her own identity, and rather than taking his. She would consider taking his name as DISHONORING his memory.
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It's obviously based on the phrase "heart of iron", which has literally been used for centuries to refer to courage and bravery...it's actually a pretty reasonable superhero name, regardless of your opinions of the character.