doesn't it actually depend on babcia and what she wants to be called.
The question is why so many allegedly Polish (but we all know that they're of uncertain peasant origin) people are using a word which is unknown in Poland.
This is how it was used in my family. My father's mother's mother was called Busha. I doubt the term is an exclusively American invention because she was from Poland, having come to America as a young adult.
You're welcome to try and find the origin. My best hypothesis is that it comes from the Ukrainian "babusha" - certainly many Ukrainians were more likely to identify as Polish than Ukrainian in those times. Nothing wrong with that, as such - but it's my theory that many "Polish" people in America were actually a mix of ethnicities, but identified as Polish as they weren't German, Hungarian or Russian.
It's certainly not a word found in any Polish dictionary at any time.