Бабушка- /ba'bu:shka/- granny, old lady
The word Babushka is often confused with the word Matreshka. Babushka means an old lady or grandmother. It initially derived from the word "baba" which signifies a woman. With an addition of suffix -ushk- the meaning has changed to an affectionate diminutive word granny.
Now it is difficult to say who started calling Matreshka (a nesting doll) babushka and why. Probably they have some similar features. Anyway it is better to differentiate these two words to be understood.