Convincing the inconvincible
I highly recommend two podcast episodes on the theme "how to argue", featuring Daryl Davis, the black friend of white supremacists. Mr. Davis is remarkably lucid in dealing with very difficult issues, and presents great tips for convincing the inconvincible.
Gather your information. Understand their position as well as you do your own. Expect to hear things you will disagree with and keep your cool.
Invite them. Have a conversation not a debate. Debates get people's guards up. Instead seek to understand them. People want to be heard and speak their mind freely without retaliation.
Look for commonality. You can find something in 5 minutes and then build on it. Look to other issues. Politics? As you find more commonality the differences matter less.
Talking is not fighting. When the talking stops the ground becomes fertile for violence (cf. Sam Harris). The longer you talk the more commonalities you can find.
Make a good first impression. Be respectful.
Don't be condescending or insulting even when you hear things you don't like. Expect opinions that are ridiculous and facts that are wrong. Keep cool.
Don't explain somebody else's movement. Let them explain it and address the points that they themselves define. Let them finish and don't cut them off even though you probably know a lot about their ideas already.
"While you are actively learning about someone else you are passively teaching them about yourself." – Daryl Davis