I too am interested in this. I have a grandfather who is otherwise thoughtful and open-minded and exceptionally intelligent, but he finds the asteroid impact theory of extinction completely impossible. Attempting to convince him has been fun in that we have a good relationship and he'll readily respond to me without turning it into a fight. If I can learn what works and what doesn't when changing his mind with available evidence, then that might work with other more difficult people.

In his case, there are a few issues:

1. Related to the other commenter's point, he's not fond of statements of absolute certainty. He likes to say, "well, we weren't there, were we?" I liked the article a lot, but I can see how it was doing really well right up until the very end, where it accelerated into story mode. Everything before that was more carefully written; "we think this because...", "it could have been this other thing, but probably not because...". For people willing to consider evidence that runs contrary to their belief system, a preponderance of evidence coupled with consideration of differing opinions is way more convincing.

2. It stretches the limits of his imagination. On human scales, imagining a global catastrophe like this can be difficult. Media, artist's depictions, CGI and so on don't seem to help much here (caveat: unless they're really, really good). Examples from other parts of the solar system do help a little.

3. He wants all of his questions answered. From that standpoint, this article is excellent. People want puzzles tidied up into nice, neat little arrangements with a bow on top. Why did some things survive and others didn't? If it just got really cold, why did the dinosaurs in the ocean die out? If it was so big, why can't we find the asteroid that did it? Weren't things already dying out anyway? What if the dinosaurs just got so big that they couldn't reproduce anymore? These are all actual questions he's had. It can be a very frustrating dialogue for scientists and the science-interested, where we learn not to expect answers to all questions right away and any answer that begins with "I don't know" is regarded as evidence contrary to the theory, but the more answers that can be provided, the better, even if they're our best guess at this time. So then you have to know your subject matter really, really well, better than the majority of hobbyists do.

4. Constant exposure helps a lot. I think people that become comfortable with lack of certainty also tend to become averse to argument and conflict, and that doesn't help their cause in the long run. You have on the one hand someone who will argue a case from a position of ideological certainty, and on the other hand someone who says "maybe" and "we don't really know" and "at this time we think" a lot. If you know nothing about some given subject, who would you tend to believe? Imagine it's someone presenting arguments for a subject you have tenuous opinions about, but not a lot of expertise or domain knowledge. I think this is why people like NDT and Nye and Dawkins are more popular in the mainstream while being somewhat distasteful among other intellectuals. They present evidence-based views with the same conviction and tonality and theatre as you might expect from a preacher.

That last point runs somewhat contrary to the first, but I think there's a happy medium where I can just send grandpa a really good article, like this one, about once a week or so, and say, "here, this does a good job of answering some of your questions, what others do you have?"