After the mother's death, she keeps going about her life as usual, goes by her original name, does all the normal things she did before. On official documents she lists her mother's name but around friends or neighbours goes by whatever they want to call her. Maybe a lot of the neighbours know what's going on, but hey, we all want to avoid taxes, right? So it goes.
And gradually she starts going by the other name instead. People nod and understand if she corrects them. And eventually everyone around her that might prove otherwise dies off, moves away, or loses contact. Decades go by. Soon everyone remembers that older spritely woman who's always been around here.
Any photos or evidence that might prove her story false, well they go missing. The ones that confirm it stick around.
This is how all legends start, really.