Eliezer_Yudkowsky comments on How to Not Lose an Argument - Less Wrong
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The first rule of persuading a negatively disposed audience - rationally or otherwise - is not to say the things they expect you to say. The expected just gets filtered out, or treated as confirmation of pre-existing beliefs regardless of its content.
Sadly, the unexpected frequently gets translated into the expected, even to the point of explicit denials of a position being ignored repeatedly in a single conversation.
This situation is especially troublesome if you've already been market as standard opponent #445, such as when you change your own mind you a new, more subtle position that is similar to your old position.
When I adjust my position in subtle ways like that, I go to pains to point it out, which is rhetorically advantageous and shows that there's a real dialectic going on.
Then say something more unexpected. There's an art to it.
How hard is it to fit a simple denial into a frame? Not hard at all.
Also, by following their arguments, trying to clarify it and understanding the pieces. Your sincere and genuine attempt to understand them in the best possible light will make them open to your point of view.
The smart Christians are some of the most logical people I've ever met. There worldview fits together like a kind of Geometry. They know that you get a completely different form of it if you substitute one axiom for another (existence of God for non-existence of God), much like Euclid's world dissolves without the parallel postulate.
Once we got to that point in our conversation, I realized that they we agreed on everything about the world except that postulate, which they were also aware of. I realized that they were neither stupid nor evil, as I had assumed before (a remarkably common, and uncivil, view that atheists have of believers). I still disagree with them. However, I was fine with leaving the conversation with both of our positions unchanged, but understanding each other better.