Green’s Dictionary of Slang

the mouth; thus as nickname of a cheeky or verbose individual .

c.1775		H. Carey Honest Yorkshire-Man 14: Saucebox! the worst is too good for you.
1835		D. Crockett in Meine Crockett Almanacks (1955) 38: I determined on trying to what virtue there was – not in stones, like the ‘old man with the young sauce-box,’ – but in a much more potent article, whiskey.
1854		Delhi Sketch Bk 1 July 82/1: ‘You’re a Sauce box any how!’.
1864, 1867, 1870		Hotten Sl. Dict.
1865		‘Mr Carlisle’ Stray Leaves 252: The time may and probably will, come, my dear Miss Saucebox [...] when you will agree with me.
1895		C. Crowe Aus. Sl. Dict. 69: Sauce Box, an impertinent young person [...] Sauce Box, [...] the mouth.
1909		J. Ware Passing Eng. of the Victorian Era.
1927		(con. 1835–40) P. Herring Bold Bendigo 253: It’s the man who wears the belt he wants to talk to, not the man who wears a head and a saucebox like Deaf Burke.
1981		M. Keane Good Behaviour 119: ‘He’s had a bit of a doings with Mr. Hubert’s horse’ ‘Is it pamper that little sauceboat?’ ‘Lucky he’s not hurt’ .