I have a feeling that most of the people reading this site already understand everything in this article, but it's a useful synopsis of common issues faced when trying to have a reasonable discussion with laypeople, and might be good to point them to if necessary.
http://thoughtcatalog.com/2011/how-to-have-a-rational-discussion/
I also want to mention how much I wish someone had shown me something like this as a teenager- I was very prone to lecture others against their will- as it might have saved me a lot of grief. I'm curious to see if these tendencies might have been common among members of this community growing up, so please comment to tell me if so (actually, please tell me even if not-no reason to encourage my own confirmation bias)!
"What would change your mind?" is a Dark Arts question. The problem with it is that the better your reasons for believing something, the less likely anything is to change your mind -- practically by definition. The person asking this question is usually presuming that the less likely anything is to change your mind, the more irrational you are being.
It's a universal counterargument to the very idea of basing beliefs on evidence. Strong evidence produces strong belief; strong belief is irrational faith. The more evidence you present, the more you demonstrate your irrrational faith. How much wiser is the one who adopts a rational faith on the basis of no evidence!