Matt Yglesias is puzzled by the term. I’ll have a go: “Intellectually honest” means you make arguments you think are true, as opposed to making the arguments you are “supposed” to make and/or avoiding making arguments that you think are true that you aren’t “supposed” to make.
Advocates, by contrast, make the best arguments they can think of for the position that they are obliged to take by their position. They are still supposed to be honest – they are not supposed to actually lie. But they are not expected to follow their own consciences with respect to the arguments they make or the positions they advance.
I’m not 100% sure that’s what the phrase is supposed to mean, but given that a substantial fraction of people who opine for a living behave as advocates, it does seem there ought to be a phrase to distinguish those who don’t. “Intellectually honest” fits the bill for me.