When you hear powerful evidence or arguments that should get you to revise your beliefs, not only do all sorts of cognitive biases fight the changes but so do the social factors of status and face saving. Perhaps I've long been a vocal proponent of X which implies Y, and you show me that Y isn't always true. It's very hard to just straight up admit "ok, I'm not a hardcore Xist anymore." There's a status loss in letting yourself be convinced.
For a long time I thought that I was stronger than this, that saving face only mattered as much as I let it matter. I wish I could freely admit when I've been convinced, but I often can't manage to. [1] Instead I'll finish a conversation defending my earlier beliefs and only later start acting on my new ones.
After a discussion where someone didn't admit to any change of mind, I'll often see them later having changed their behavior. So now if I'm trying to persuade someone I don't focus on securing verbal agreement. Instead I just try to be as convincing as possible, and notice if they come around later. [2]
(I also posted this on my blog)
[1] This is not a helpful trait: I'd like other people to let me know when I'm wrong or when they have evidence I'm not considering, but if they never get the satisfaction of knowing they've convinced me they may just feel like they've wasted their time, and not try in the future. So I'm working on it.
[2] Keeping people from feeling personally invested in one side or the other of an argument is probably also helpful: I understand discussions are much more likely to convince bystanders than participants.
A sneaky move would be to do something like: "I don't know, umm yeah.. thinking about it from this viewpoint it would be interesting if .. " and suddenly listing previously not said(and usually more tangential) arguments going against your previous argument and supporting your pal's statement. The point is that you somewhat start to act like a judge of your own discussion, accumulating even more status retaining control and the image of a reasonable man.
Completely steal the show. Although it usually works only with one person dialogues, if others are present they would assure your puzzled pal that the credit indeed belongs to him ,)
Another good flip(or reframing) would be "ohh yeah let's continue just for the sake of the argument, to see how far this point could be derived" than defend it with your original intention. If works you have it, if not it's not attached to you anymore.
But sometimes you just feel the insignificance to the universe and let it go with a smile.