Sniffles:
From my informal understanding of the expression of someone having a "thick-face" or "thin-face"...let me use it in a context so I can better explain and you can understand it as well. Yes, it sort of does have to do with social reputation.
Let's take for example, this guy is pursuing this girl, but she had already express disinterest in him. However, he continues to follow her and a bunch of her friends around because they go to the same school or whatever. No matter what her friends or even what she would say to him--insult him or mock him, so that he'd understand he's not wanted and move on. But he continues to pursue her. They would refer this guy as a guy with a "thick-skinned face". He shows no shame in being rejected or disliked. I mean, these insulting comments are probably said in a indirect way but it is said so he can hear them, so even after hearing them, he does not seem to care, then he's quite thick faced.
Therefore, if a character would say "I'll thicken my face", it would be said in a very cheeky manner. As it is a cheeky comment, no one would say that unless it is in a joke.
Now a thin-skinned face would refer to someone who is easily affected and easily embarrassed by people's comments or looks upon them. In other words, a very sensitive drama-queen or king.