In the suggestions, an old woman was a possible translation. I read a story (in English) in which Ivan called Yaga Baba 'babushka'. I guess in Russian, you call every old woman 'babushka'.
That must have been an old fairytale. This character probably was meant to be shown a goodhearted hero speaking to a hag in such manner. Some time ago it was a warm respectful way to talk to an old wise mother(or grandmother I guess) figure. Nowadays it's the opposite, even though people using it usually mean well, it's inappropriately condescending.
Finally! You are the only person I’ve encountered on this whole forum to have mentioned the Slav superstar