I was thinking of something along these lines the other day, while trying to go to sleep. More or less, I realized something: Conflict, in D&D, is inherently characterized in terms of physical stats.
What if we turned that around? What if the physical stats only had ancillary or special-case application to conflict, while Intelligence, Charisma, and Wisdom provided the central heft?
From there, I began to wonder. What stat would make most sense as a mental equivalent of HP? I often hear Wisdom described as such, but to be honest I feel like Charisma is the better choice. It represents
force of personality; a low-Cha character is better-represented as
meek, or possibly
solitary, someone who is easily swayed to the ideas of others. Then, to avoid the "god stat" problem that (IMO) is so well-known in regular D&D, pick one of the other stats to be your mental equivalent of AC, and the last stat is your mental "initiative." Again, personally, I feel Wisdom is the better choice for the "mental AC" (you easily catch flawed arguments, and know many "practical-thinking" techniques), and Intelligence is perfect for "mental initiative" (it's specifically about how
quickly you process information and how well you retain it afterward).
To distinguish mental conflict from physical, we could call the mental equivalent of HP "resilience points," RP (yes, the pun is intended; if you don't like it, you can use "determination points," DP, instead). I like calling the mental equivalent of initiative "acumen," though other names might be better. And mental AC could be, say, "wit class/WC" (not so fond of that one, would welcome suggestions).
Then, each class could have some particular thing that it's good at; you could perhaps even make a whole mechanical angle, an "argument class" or whatever (trying to avoid the...well, rather political-sounding "social class"

) that would provide your mechanics for such "mental" conflicts. Some could be deep and incisive question-askers like Socrates (effectively "damage-dealers"--they don't hold a position themselves, and thus may be easily forced away, but they can really HURT another perosn's position), some could be highly difficult to convince, etc. Although my thinking is heavily influenced by roles (because I find that kind of design elegant), you could totally do other stuff with it too.