During that time I attempted to fly around the sim and figure out if it was a client or server problem. I've attached a few screenshots of the sim stats and the frame console when the problem struck.
The first pic shows this problem at its worst. The frame console indicates massive UpdateGeom usage -- so much that the video lag actually affected my network latency. (Every time my framerate dropped, my ping would spike.) The second pic is similar, with me in a different part of the sim while wearing a Ruth avatar and no attachments but a flight enhancer.
I set my video options to the minimum to see if it was my client. The only one that made any difference was, of all things, Shiny Objects! With shiny off, this problem vanished completely. With it on... bam. Framerate suckage.
The third screenshot shows one other interesting piece of data: the "Verts Relit" statistic. It was the only stat in that window that consistently spiked along with the UpdateGeom numbers in the frame console.
I have no idea what would cause this problem. I haven't changed my video settings or drivers in at least a month. The build I was sitting in hasn't changed for almost as long. My friend and I were just talking, no scripting or shooting guns or anything of the sort. The build itself has a fair bit of shiny objects, but when I rezzed a copy of it in one of the sandboxes... no lag. I don't think it was the build.

I suppose I could just turn off shiny, but that wouldn't really tell me what caused the problem, especially since it affected my friend's client, too.