The noise was coming from the rattling of that red-orange thing in the picture. The noise went away after I shook it out. Hopefully it's not an important part! I'm amazed the motor has been able to run all that time with it floating around inside.
I actually ended up removing some of the grease that was in the gear box. The whole thing was filled with it. I should probably go back and do a thorough cleaning and re-greasing instead someday. I do worry about the opening to the bobbin area and all that stuff that could fall in from there. I wonder if the excessive amount of grease was meant to provide a barrier of sorts. It was actually green-colored and looked a bit like an encasing of jello or something. It was quite a sight.