Randy Falcon grew up in the French-speaking region of southwest Louisiana listening to the Beatles, the Rolling Stones and the Dave Clark Five. He had little interest in his family’s Cajun music royalty. A second cousin, Joe Falcon, had Cajun music’s first commercial recording, “Allons a Lafayette,” in 1928.

Still, Falcon was intrigued by the 10-button accordions that fueled Cajun songs. Early squeezeboxes that had made their way to Louisiana from Germany before World War II had the supply...