Green’s Dictionary of Slang

(orig. US) a phr. of encouragement (esp. in response to a request to borrow something), go ahead, ‘feel free’, ‘help yourself’, ‘make yourself at home’ etc.

1952		N.Y. Herald Trib. 29 Oct. 25/1: One expression that has overflowed Hollywood but probably originated here and is still in wide usage: ‘Be my guest.’ This can be used almost anywhere.
1959		W. Brown Teen-Age Mafia 86: ‘What about another cup of coffee?’ ‘Sure. Why not? Be my guest.’.
1960		J.D. Macdonald Slam the Big Door (1961) 6: Eat well [...] Be my guest.
1971		N. Smith Gumshoe (1998) 98: For you, Eddie, be my guest.
1992		R. Puxley Cockney Rabbit 71: Georgie Best 1. guest [...] ‘Please, be my Georgie Best.’.
2002		J. Lerner You Got Nothing Coming 1: ‘May I read it?’ ‘Do I have a choice?’ ‘I’m afraid not.’ ‘Then be my guest.’.