There's been a lot of talk about "framing", but I think there's a step that has be be taken before we can even start developing the right "frame" to convince people.
It seems pretty simple, but literalists that we are, we often overlook the fact that what people say is not what they really mean. So, if you address what a person is saying at face value, you may well not get anywhere, and in fact, the person arguing with you starts to seem more and more irrational because you are not getting to what they really mean.
Let me give some examples, including personal examples...
One of the most obvious is abortion. There are plenty of people who talk about how horrible abortion is, but what they really mean is that sex, especially out of wedlock, is what they really find offensive. Thus reducing abortions through birth control and sex education really doesn't do anything for them--it might make them even more angry, in fact, because people would then be "getting away" with having sex out of wedlock without the evidence of pregnancy to show how sinful they are.
My personal example is the death penalty. Unlike most of you folks here (but like the majority of the country), I am in favor of the death penalty. Mind you, I would probably not favor it in most situations, but my stance is that I wouldn't want to give up that option altogether. Mr. JMS, bleeding heart liberal that he is, has browbeaten me with every liberal argument on this subject, even to the point where when in defending myself I have seemed a little irrational, but I haven't budged. And I'll let you in on why. You'd first have to understand why I favor the death penalty. To my mind, there are some crimes--not many--but a very few--that are so dire that I want the perpetrator to be punished as forcefully as possible. And to me, death really is the ultimate punishment, because as long as there's life, there are possibilities. I don't buy the idea that a guy sitting around thinking about his crime is undergoing a worse punishment than death. Maybe torture and THEN death would be worse, but try getting a majority of people in this country to go along with that. So maybe I'm bloodthirsty--probably moreso than most around here. But if you don't get that about me, you can't get at why I would favor the death penalty, and you certainly can't even start to figure out how to get me to change my mind.
Many of the "whys" of why people take certain positions really do reside more in an "id" or something that's below the rational level. They reside in things like xenophobia, racism, elitism, misogyny--primal places that we cover up with rational arguments. Englightenment types are constantly surprised when their rational arguments don't work. But, to me rational arguments would only work if they are in line with the deeper undercurrents, OR if the topic discussed is not of any particular importance to the arguer.
So as we go forth, for every frustrating, irrational, obstinate viewpoint you meet, try to dig a little deeper and figure out what's really going on.