Yeah, it's interesting if you're into etymology or language in general. Their claim is basically "it's our last name, therefore our ancestors must have named it after themselves", but the term and its resultant surname can refer to a bunch of different things and there's little evidence suggesting their claim is true.

Broccolo is the Italian term for the brassica flower crest and is the diminutive form of brocco, meaning sprout in a botanical context. Broccoli as a surname can mean their ancestors were broccoli farmers, military or related as a brocco is the center protrusion of a shield, carpenters as it can refer to a type of nail, or even arborists as it can be the stump left after cutting off a tree limb. Italian is a very old language, the associations get wild and sometimes don't make a ton of sense.

Broccolo is also a silly term to call someone an idiot.