'baba' originates from Slavic languages (Polish/Proto-Slavic), specifically the Polish word 'baba', where 'baba' meant 'old woman' or 'grandmother'.

'baba' passed into French as 'baba' (referring to a cake) and entered English from French and Polish; the modern English use for the cake (and the quoted familial sense) comes from these routes.

Initially, it meant 'old woman' or 'grandmother'; over time it was applied to a type of cake ('baba au rhum') and retained its familial sense in quoted borrowings.