Hanna , I don't think legal rights are what Rielle Hunter's after. (Although I can tell you that the relationship between the child and John Edwards may be covered under North Carolina General Statutes Chapter 49: Bastardy.) Even if John Edwards were to die intestate, an acknowledgement wouldn't change much. Given the circumstances, I'd imagine any probate court would agree to DNA testing in the unlikely event the Edwards estate were to need to be divided up according to the intestacy rules, or that another person made an unclear reference-to, say "the children of John Edwards" in a will. Beyond inheritance, it's hard to see what the child gains-the right to be in on a decision about whether to pull the plug on John Edwards, if circumstances required? The right to pen a tell-all memoir without fear of legal repercussions?
I agree with Emily that what Rielle Hunter seems to be after is less legal than societal. We could give her the benefit of the doubt, now that her daughter is getting older, and suggest that she's trying to normalize the situation for the girl as best as she can. After all, a baby can seem like a pawn-easy to move around and produce at opportune moments. An eighteen-month-old is sitting up, grabbing at things, and probably saying a few words. Maybe Ms. Hunter is looking ahead to the questions her daughter will ask, and hoping that proximity to John Edwards--if he is the baby's father--and his family might mean that the girl can grow up with her unusual parentage no more than just another fact of her life, at least for a while.
Or maybe she just wants to stick it to him.