Thanks for the feedback guys. The Fixer's right, it's not an issue of the cable pinging against the top tube (I've got donuts up there anyhow). And, the ferrule is not contacting the frame in any way that I can discern.
After looking at it more closely, it appears that the cable is actually rubbing up against the housing when it is moved from side to side (i.e. during a turn). It almost feels like there is a frayed end in the housing that it catches on, then jumps off. I can feel it when I grasp the cable between my fingers and move it back and forth at the housing exit. That may be the root of it all.
While googling about this, I found a tidbit from jimlangley.net:
Cable housing sections can click as you turn the handlebars to steer.
Lubricate the point where the housing ends enter the frame stops. Often, the end of the cable housing has a metal or plastic cap (called a "ferrule"). This can move when you turn the bars and make clicking noises. The end of the housing can also move inside the ferrule and make noise. Usually, applying a few drops of medium-weight oil to the ferrule, the housing and the frame stop will stop the clicking.
I'm going to try this when I get home this evening. But I have a feeling the housing might need to be replaced or at least smoothed out inside, since I assume there should be no rough edges. I imagine that when the bike was assembled the housing wasn't cut smoothly (or it wore away and frayed during usage).