In a slinky dress on a balmy May night, "The Bachelor" star Hannah Godwin was making the rounds at a Brooklyn rooftop party with a cocktail in hand: No, not a martini — not even an espresso martini.

Blue raspberry is one of those flavors you instantly recognize: It’s bright, tangy and unapologetically electric-blue. You’ve probably sipped it in a slushie, chewed it in a gummy, or powered through ...