I've never before heard that you're not supposed to lubricate the star
wheel mechanism and I'm wondering where this idea came from?  I'll have
to double check, but I'm pretty sure that Regina's instructions clearly
stated how you're supposed to lubricate it.  In any event, it's steel
and can rust so, by all means, do lubricate it.

I use clock oil in a lubricating tube with a long thin point to ensure
that the oil is coming out exactly where I want it, and not too much.
The lubricating tube and the oil are available from Nancy Fratti or
from clock repair supply stores.  Just a small amount across the axis
is good here.

I would definitely not want to use graphite or "dry lube" here, and
certainly not an oil that may gum up or leave residue.

Eli Shahar