While using a Dremel cutoff wheel to shape a lathe bit, the flexible shaft sounded not quite noisy, if slightly less smooth than before, so easing some oil into the drive shaft might be a Good Idea. The springy shaft slides out of the motor end without disassembly, but, just for completeness, I took a look inside the handpiece:
Before removing the two obvious screws holding the handpiece together, remember to remove the steel ring at each end. The tail ring is obvious:
The nose ring wasn’t where I expected it, but released easily after the obvious mistake revealed itself:
The steel shaft spun freely in its bearings and the matching end of the flex drive shaft had plenty of grease:
So I just reassembled everything in reverse order. The trick is to line up the existing indentations in the outer sheath with the bumps inside the handpiece shell:
After all that, spreading a few drops of high-speed spindle oil along the spring drive shaft seemed appropriate.
Didn’t make the least bit of difference to the sound, but I feel better.