Start by picking your audience: Are you aiming to influence people at your Thanksgiving table who are mostly listening to you and another partisan debate? Or are you hoping to convince the person you typically argue with?
Choose specific goals: Instead of trying to convince someone to do an about face on everything they believe, pick something smaller.
Political conversations are like two lawyers arguing with each other, with no judge present. Give that up: avoid scoring points or aiming to make anyone look stupid. There are lots of ways to change the game:
What can you exchange? Fairness is a value held by both the left and right. If you want someone to listen to you, listen to them first, repeating back and making it clear you understand their points (try active listening). Let them talk themselves out and feel heard, let it be so unbalanced that even they feel it is your turn to talk. Get good enough at understanding their point of view that you can say it, including the underlying positive values beneath it. Help them feel heard -- not being heard may be a deep motivation for turning to angry politicians who seem to speak their fears. If their political goal is to “score points," give them some. Find things they believe in that you believe in too, and say you agree. See if you can switch to a fairness paradigm, you will listen and agree with some of their points, and so they owe you at least listening.
Anger Politics: If you think the right-wing machine is dividing us and is powered by anger, then seek more connection. Offer to read a book together that's not on hot-button issues. "You say I don't know enough history, but before the next holiday I'd be open to reading a biography of Thomas Jefferson, or that guy who wrote Common Sense that helped the American Revolution catch fire."
Students: Giving high school or college age students suggestions for paper topics can be very powerful. "What happened with Nixon, Chile and Pinochet? Maybe that's a good paper topic?" Doing the research and writing a history paper on the CIA and Iran is likely to convince someone far more than being told. Let them find out for themselves.
Draw new boundaries. For solid Democrats or leftists who believe that the Republican Party has chosen hate-politics since the Southern Strategy, and want to make them pay for it, this Thanksgiving may stand out. The normal path in Republican politics is to come back together after the primaries: social conservatives and libertarians will circle their wagons against liberals. But the anger politics may have gone too far. The best way to nudge it over the edge is not to argue, but to let conservatives speak about the things that divide them. Ask what they think about Trump viewing Iraq, whether hedge fund managers are paper-pushers who should be taxed more, or ask about gun control for the terrorism watch list. You don't need an opinion -- if the Republican machine wants to build a political movement off Fox news and anger with anyone who disagrees, just let them turn it on themselves now. There may be other boundaries where you can agree with your relatives on a non-hot-button issue — erasing some divisions between left and right is far more worthwhile than winning a dinner argument.
Ask questions. With Bernie Sanders trying to make us more like Denmark, you might ask Pro-Life relatives what the abortion rate is in Denmark, how it compares with the Red States. Even if you know the answer, don't answer, questions are more powerful than answers. If you have an audience, ask if Syria was ever colonized and by who, was Iran ever a democracy, was Saddam Hussein ever an ally? Blasting people with facts just becomes a game, where you are the lawyer for your side, and they will become lawyers for their side. But if they don't know the answer, they may look it up -- students need school papers, and studying the history If you believe your side is right, think about the underlying facts that brought you to that conclusion, and ask questions that will lead people to those facts. Trust people to figure it out for themselves, it's much more powerful that way.
Ask real questions. Do you know Syria's history? I don't know much. That could be an interesting thing to look into together, instead of arguing where our opinions are set. Or ask questions about motivation, what feels really important. Ask how old they were when they developed their political values, and what led to it. The more real your questions, where you don't know the answers, the farther you get from the world of opposing lawyers in unending battle.
Avoid strategies and seek values. It's hard to have a deep conversation about minimum wage, because liberals and conservatives think it will have different effects. Seek out the values underneath, ask questions that are not about today's politics: what will happen if robots can do all the work? Buried fears make for the worst politics. Real conversations, even if you don't agree, can help reconnect our politics, and make grand-standing less acceptable. You might check out Jonathan Haidt's studies of the different values held by liberals and conservatives. Explore whether these value differences hold true at your table.
Avoid stale anger. Whether you are left or right, don't get angry in the abstract. Don't get angry with conservatives and therefore with your conservative uncle, keep Thanksgiving personal. If you have to get angry, if you are being insulted, yelled at or ignored, get angry only at the person doing it, and only for what they are doing. And again, questions can be powerful: ask people what is motivating them, why are they angry?
Thanks-giving. One powerful technique for getting near the roots of our politics, without actually being political, is giving thanks for the things in front of us. Give thanks to whoever picked the cranberries, to the farmers and laborers, to the cooks. Don't overdo it: if you can get everyone to realize that they already know that the cranberries were picked by people who've left their families to work as undocumented and underpaid on farms in the US, that's better left unsaid, where it can't turn into a shallow political argument.