Activism - Changing Someone's Convictions
Here's another story showing how someone's mind can be changed and what types of arguments and evidence might land. (This is obviously not an endorsement of his particular conclusions or positions, though I like what he says about Islam and free speech.)
Incidentally, I think the task for Objectivists is both easier and harder for the same reason. Easier because every argument we offer is integrated and coherent and so over time the logic of the case builds. Harder because it demands that someone unlearn many of their conclusions and then replace their whole worldview (if they have one) or, if they don't have a worldview, to learn to approach particular issues as related and then to systematize them.
And given this, I think it's hard to tell how far along someone might be to being generally persuaded, hence it's always worthwhile (given your time and context) to continue trying.