I always use the default name anyway, so I wouldn't really mind if there was no option to change the MC's name. But if the MC has a really strong personality or is so different from me that I can't relate to her, then that's a problem.
I know most otome games try to make the MC's personality pretty neutral and unobtrusive so that as many players as possible can project themselves onto her, but even then, if she says or does something that I would absolutely never say or do, it can completely break my immersion.
So for me, the name isn't really the important part. What matters most is whether I can actually immerse myself in the MC.