。
My mother was busy preparing dinner.
Literal
Mother [topic-は] dinner [possessive-の] preparation [object-を] do [nominalizer-の] [with-に] was-busy
Nominalized action + に忙しい — の wraps 準備をする into a noun phrase, and に marks it as what one is busy with. 準備をする and 用意をする are virtually interchangeable in this 'preparing dinner' context.