In the ‘50s and ‘60s Lucy from Peanuts was frequently referred to as a fuss-budget. The word’s origins are unknown. “Budge” comes from the French word “bouger” which means to bubble or boil so it’s possible the word comes from the notion of a person that fusses, worries, and complains to the point of boiling over. The word is related, obviously, to “fuss-pot”. And of course pots boil over...

Alternatively “budget” comes from the French word “bougette” which means “leather bag”. Another related fuss-x (cool band name alert!) term is “fuss-box”. So possibly budget/box/pot all mean to imply the person is nothing but a container of fuss and worry.