The way Cedryl Ballou sees it, he had no choice but to pick up the accordion and make his mark in zydeco music. His paternal grandfather, Gulf Coast blues guitarist Classie Ballou had backed up Boozoo Chavis on one of the first true zydeco records, “Paper In My Shoe,” while his maternal step-grandfather Rockin’ Sidney Simien had revolutionized the genre in the ’80s and kick started its current popularity with his gargantuan hit “My Toot Toot.”

“After my grandfather had passed—Sidney—I deci...