I don't practice in New Jersey, so I don't know all of the applicable laws and regulations, but here are a few suggestions.
First, your sister should check to see what the laws are regarding unemployment. Where I live, if you are terminated for cause, you are not eligible to collect unemployment. And although your neighbor was fired for cause and is collecting unemployment, this does not necessarily means that that is what the law is. Some employers simply choose not to continue fighting after they lose an appeal, for whatever their reason may be.
Second, your sister certainly should request that an HR rep be present at any meetings with the supervisor when your sister's progress on the PiP is discussed. (And just FYI, had she not signed the PIP, she likely would have been terminated for that alone.). At any meeting, your sister should document what is discussed, and ask for specific examples about how she is or is not meeting the goals. If possible, she could try emailing her write up of the meeting to her supervisor and HR, because this gives her documentation should she need it later.
Additionally, if she feels that she is being unfairly targeted, she should discuss the issue with HR. Just out of curiosity, is she a member or a protected class other than being female? Does she have and specific instances of other employees not in that protected group that are being treated more favorably in the same or similar circumstances? If so, she should document these as well.
Overall, if she thinks (either with the aid of HR or not), that she can meet the expectations of the PIP, then she should certainly stick it out (although I would most certainly continue searching for jobs elsewhere). However, if she does get fired, many future employment applications may ask whether she has ever been terminated, in which case she would have to answer yes. But obviously, as others have stated, that is not a complete deterrent to getting a job.
But definitely tell your sister to document what is going on (preferably by email or some such method where it is possible to ascertain when the document was created, so it doesn't look like it was created after the fact). I don't know if New Jersey has a wrongful termination claim or the only wrongful termination claims stem from discriminatory terminations (which is how it is where I live), but if she thinks that she was wrongfully terminated, she should speak with an employment discrimination lawyer.
Oh, and again, I don't know how it works in New Jersey, but in some states, it is difficult, if not impossible, to bring a claim for a discriminatory termination if you quit your job. She would then likely have to make a constructive discharge argument, which is very difficult to win. She would need to check on that (or speak with an employment attorney) to find that out as well before she makes any decision.
And again, I'm not licensed in New Jersey so don't take everything I say as gold.