How would you tell, what would this "new person" theory predict differently then the old person theory?

How plausible that is depends, I think, on what you mean by "person".

They are probably thinking of person as a soul. Kickstarting the brain might give you different thoughts or some other trivial thing, but the ghost should be the same, right?

I first wrote the quote below first. I'm not so sure now. Maybe some personality is basically damage.

No, I don't think you get a new person. Any damage or distortion won't project onto another person in mindspace, but just a broken or distorted version of you. There may be less of you, but not more of someone else.