It doesn't seem to be in current use, but as recently as the 1960's, people would "go for a bit of how's your father" which meant going out in search of casual sex. I get the idea that "How's your father?" itself was used as a euphemistic chat-up line, similar to "Come up and see my etchings," or the current favourite, "Netflix and chill".
I understand the social need to use a harmless pleasantry as a deniable proposition, but why this phrase in particular? What was its origin, and when?