According to some (unverifiable and untraceable) sources, it seems that Queen Margherita di Savoia was a very chatty person ("chiacchierona"), and she once supposedly asked to one of her favorite cooks during her visits in Naples some sweet food to eat while spending time with her guests.
Somebody then named them "chiacchiere" in her honor (?), but some say that she choose it since they were intended to be eaten while chatting.
Some say that that cook was probably Raffaele Esposito, which is the one who "invented" the most famous type of pizza, Pizza Margherita, that was named (again, supposedly) to honor the Queen.
"Chiacchiere" is a term that is sometimes used as synonym of "bugie" (not literally "lie", but mostly meaning unreliable news, gossip), which could be the reason for this alternate naming in some northern regions of Italy like Piedmont and Liguria.
Unfortunately, when food names are not clearly related to their shape or preparation, it's almost impossible to find the real origin behind them, and explanations are usually just... chiacchiere.
Some references (not reliable sources, though):