Question

The best way to win an argument with a woman is to hit her over the head with a new mink coat meaning

Answer 1

There is a humorous meaning for the sentence.

First, people think that it is impossible to win an argument with women as they are talkative and try to win an argument in any way.

Second, women love cool clothes.

A mink coat is the best quality coat made of soft fur. It is originated from the English.

So, the sentence means that if a woman is to be kept quiet, she should be presented with a new mink coat which literally means, by gifting her the best clothes she would stop the argument.