I've been chewing on this question for a while.

This WP article could serve as a starting point - though it looks a little daunting. It makes a lot of a Stephen Toulmin's "six elements of an argument" - I see that Toulmin hasn't been discussed on LW so far. I'll see if I can get some info, summarize and evaluate the usefulness of that framework.

A proposal in line with M&S would be: a good argument is one that causes your interlocutor to accept your conclusion. A good counter-argument is one that justifies your rejecting your interlocutor's conclusion. This conforms to the hypothesis that reason serves argument, and that its twin functions are to help us convince others and to resist being convinced.

I'm also wondering about a "memetic theory of argumentation", where an argument spreads by virtue of convincing others, and mutates to become more convincing. Our "rules" for correct argumentation are themselves but memetic fragments that "ally" with others to increase their force of conviction. For instance, "we should reject ad hominem arguments" is a meta-argument which, if we expect that our interlocutors are likely to use it to reject our conclusions, we will avoid using for fear of making a poor initial argument. In this manner we might expect to see an overall increase in the "fitness" of arguments as a consequence of the underlying arms race.

We should also be careful to distinguish conversation from argument, I see the former as serving an entirely different purpose.