Green’s Dictionary of Slang

1. to shock.

1978

Eble Campus Sl. Apr. 1: blow (somebody) out, blown out – to embarrass [...] to shock by doing something socially gauche. When she burped at the party, she blew him out.

2. to ignore, to dismiss, to reject; for one of a couple to abandon the relationship.

1970		G.F. Newman Sir, You Bastard 202: What’ll you do, blow him out?
1982		P. Redmond Tucker and Co 106: You’ll have to get it together and ask her then, won’t you? Even if she does blow you out, you’ll have sorted it one way or the other.
1999		K. Sampson Powder 346: He was going to call Celeste when he got to Boston, tell her everything, face the music and, if she blew him out, it would be no less than he deserved.
2000		Indep. Rev. 24 May 7: If I have to blow these guys out, it’s just as scary as blowing my missus out, or my mum and dad out.