Theres only one good way to change someones mind over something that they have become entrenched about - for example politics, but anything where the reaction is a no rather than a what.

And thats to listen to everything they say, and ask the right question at the right time, a gentle interjection, something that nudges them to question something themselves. At somepoint they might even ask you about you perspective, and you need to give the right kind of answer.

Its slow and painful, and for big things it takes years and years of work to get someone to change. But its the only way ive found to truly work.