Monster Alice and I are fans of the Freakonomics radio show and books. Dubner and Levitt are consistently thought-provoking (and entertaining), although I don't always agree with their conclusions- there's a bit of "when all you have is a hammer" approach to some of their work...
That being said, they do come up with some things that are worth remembering. I just finished reading Think Like a Freak, and there was a chapter in it titled "How to Persuade People Who Don't Want To Be Persuaded". It seemed like a useful set of guidelines in this day and age. Note: not everyone can be persuaded. You don't have to persuade people who can be persuaded. Not every situation is worth your time and energy- choose your fights.
Anyway, briefly summarized:
1. Persuasion is hard. A person's deepest-rooted beliefs tend to be based on ideology, peer pressure, and other non-rational processes. It's very hard to get at them rationally. It doesn't do you any good to have the best reasoned and most factual argument in the world if it doesn't resonate for the recipient.
2. Don't pretend your argument is perfect. The recipient will always be able to poke holes in it... and they may be as smart as you are or even smarter. So be aware of the flaws and the potential unintended consequences.
3. Acknowledge the strengths of the recipient's argument. You can find things to strengthen your argument in it... and it also teaches you about your blind spots, your areas where you are being non-rational. Also, if you ignore someone's argument, why should they engage with you at all?
4. Keep the insults to yourself. This follows from the above... name-calling makes enemies, not allies.
5. Tell true stories. Not simple anecdotes, but frame your facts in a compelling narrative- and resist the temptation to use hyperbole and unsupported claims; check your facts and make sure your news isn't fake. Stories hold someone's attention far better than dry statistics.
As I said, not everyone can be persuaded... but it's good to have an approach in mind.
- How to Persuade People Who Don't Want To Be Persuaded