I took four years of Russian in high school, and the instructor always pronounced "babushka" with the accent on the first syllable. And it does mean grandmother -- and dyedushka was grandfather.

He explained that "babushka" became popular as a name for a piece of women's clothing because it was associated with older, grandmotherly types. I have no idea if that's true, but it seemed to make sense at the time.

How and why the accent was shifted to the second syllable, I have no idea.

It is strange indeed as most English words have accent on the first syllable anyway... so who knows how that happen...