Diagnosis

It appears that the problem was too much play in the rotor shaft: when it sits on the ground, the shaft drops the bottom (large) gear down by about 1mm, causing it to partially disengage from the motor (small) gear.

The easiest way to fix this is to reduce the slack in the shaft. The rotor itself and the gear seem pretty firmly glued on, so I didn't fancy my chances of pushing them together any more without breaking the 'copter. In this picture, you can see the extra space between the bottom of the rotor and the white nylon spacer which is causing the vertical movement in the shaft.

Making a spacer

Since I didn't have any way to remove the rotor without breaking something, I decided I needed some kind of clip washer to go between the rotor and the spacer. When the helecopter is operating, the spacer will sit on the bearings and the rotor will rise up, leaving all the slack between them (as indicated above by the arrow). This then is the best place to add an extra spacer.

I started out by stripping a small piece of wire from a twist tie. This length was actually a bit short, as you will see in the next image.

I folded the wire in half, leaving a loop which will form the washer. The next step was to hook this loop around the rotor shaft in the space, and twist it closed to form a ring which won't slip off.

Finally, I trimmed the extra wire with wire clippers as close to the shaft as possible.

This left me with an extra loop of wire in the slack space, which pulls the gear up and into the right position when the 'copter is sat on the ground.

All fixed

You should just be able to make out the wire washer in the enlarged version of the above image.

With this modification in place, the grating noise is gone, and take-offs are much smoother. I'm having much more luck getting the helecopter stable after launch, so I count this as a success.