The Malbim explains this to not be a literal discussion about what her name is, but about how to relate to her state of poverty.
The people, upon seeing her returning from Moav, commented "Is this Naomi who was so rich when she left?" i.e. it's a shock how poor she is now, because of how rich she was when she left!
She responded, "I don't have to have been Naomi to be shocked at how destitute I am. Even had I been of average wealth before, now I am so poor that it is noteworthy without contrast to my prior status."
Secondly, she was commenting that her previous wealth was not really to be viewed as a state of blessing, as she was now realizing retroactively that her prior wealth was only to create a bigger fall when she fell, so her "being Naomi" was not really a state of "Naomi" - it was a hidden set up for the fall.