Pick one (or several) of the following. I used specific examples, therefore anything similar still counts.
1. You have a friendly new acquaintance who is pretty much an average person. He is a theist and doesn't believe Evolution, you have already had a polite debate about that. Convince him to believe in the truth*.
2. One of your friends is very deeply religious - he has devoted his life to already invested a lot of it in religion. Unexpectedly, he is also highly rational (as a personality) and very intelligent, he studies a technical degree (enjoys it), he has read books about critical thinking (he even knows a little about biases) and he says that he will stop believing in religion if you disprove it. Debating with him so far didn't help (also he isn't too good - he isn't aware of expected value and such ideas). For his own good, convince him to change his mind in the direction of the truth. He is wasting a huge potential and that's not only bad for him, but also for humanity. Also, he will feel more comfortable in his new, more sensible beliefs.
3. Your brother dislikes you because of his impression of you that was created several years ago and wasn't updated to reflect the changes in your personality. You easily make impressions to other people that are vastly different from his impression of you. Change his impression, so that he sees you truthfully.
[I have removed 4., because it wasn't about changing the mind of someone who isn't a rationalist, but about coming up with a good psychological mechanism - it deserves an entirely new thread; I suspect that 3 might be too different from 1 and 2, but it's too late to make a so big change to the thread]
I know at least one person for each category. And I haven't been able to change nobody's mind. Have you succeeded in a similar situation? Regardless of whether you have, what strategies do you think would be winning in the 4 situations? If some of them sounds good, I might even try them out and share the results. I'm especially curious about how to approach in #3, because if there is a way, it would come from low-level psychology, which is something I adore.
So, the aim of this thread is for the participants to try and change someone's mind and then tell the story.
(also, I'm willing to accept ideas of other templates for classical situations similar to those, in fact I think I had one or two more ideas, but I can't seem to recall them)
*Needless to say, if at any point, anyone proves to you that his direction is in fact the truth, it would be better to change yourself in that direction instead, but that's outside of the scope of the thread.