> Have you ever actually gotten that to work? Have you personally ever changed a single persons mind?
I know the question wasn't directed at me, and it's not with respect to the proposed Muslim ban, but I have changed the mind of some of the voters from this general background: specifically, would-be Roy Moore voters. I convinced a good handful of them not to vote for him[1]. And as far as I know, they stayed home.
The key is to be extremely patient, honest, and respectful. You can't use the typical Twitter technique of beating them over the head with opinions, or even the blunt HN technique of overwhelming them with facts. You have to first try to understand their viewpoint, and where they are coming from. Doesn't mean you have to agree with it, just understand it. And the best way to do that is to start asking lots of respectful questions, and depending on the person, sharing your own background. Then, start interspersing harder questions about the subject in question, like Roy Moore in this case. Don't push too hard, but ask hard questions in a respectful way.
Of course, in these situations, you have to be open to having your opinion changed, as well.
It's not a very eloquent summary of what I've done in this situations, but it's the best I've got tonight.
1. Perhaps the hardest part about this is differentiating between propaganda sock puppet accounts, whom you will likely never convince, and actual people who happen to believe some pretty crazy things. Once you've identified real, genuine fellow citizens behind whatever pseudonymous social media account, you're half-way there.