The fact that they're dead hardly seems relevant. Right now today you can give my birth state $20 and pinky promise that you're me, and they'll mail my birth certificate to any shady shell company you choose. You can bootstrap that document into the rest of my identity. If that's the standard of evidence for impersonating living people I don't know why we'd go through so many hoops to make impersonating dead people hard.
But in this case, it seems there is no procedure at all. She is never asked for evidence A, B, and C. She's essentially asking "this is an error, what can I do to fix it?" and the answer is essentially "nothing."
As the in-between poster mentioned, he routinely does it with his kids. I presume he's in a country where this is easier then. That's why I mentioned France vs. being somewhere else.
Here in Canada I could believe for example that this might be easier (though I have nothing but conjecture). I say that because I do know that getting your papers for the very first time "all" you need is for two persons that are already Canadian citizens to affidavit that they've known you for at least 2 years (IIRC) and that they think you are who you claim you are. On top of like your birth certificate. But thats not necessarily Canadian and not sure how much that can or will be checked.
I know from some colleague who is Vietnamese, born somewhere in China while fleeing, who had no birth certificate and doesn't know his actual birth day (just roughly the year), never mind where he was born exactly. When he wanted to do something that required a birth certificate a Chinese co-worker offered to "talk to a guy" and "get him a birth certificate".
I presume France lies on the other end of the spectrum :)
The problem is that we're really poorly equipped to handle problems of identity. It's hard to prove that you are you, absent documentation. Documentation usually issued by a government, or by a private agency on behalf of a government. And if that government thinks you're dead, well, all that documentation is worth about as much as the paper it's printed on.
Short of, say, a dental, medical or DNA match, then yeah, this is a really hard procedural fix.