Might've been the other way around, his family was named after the procedure:
The term has also been explained as deriving from the verb caedere, 'to cut', with children delivered this way referred to as caesones. Pliny the Elder refers to a certain Julius Caesar (an ancestor of the famous Roman statesman) as ab utero caeso, 'cut from the womb' giving this as an explanation for the cognomen Caesar which was then carried by his descendants.