Bob Pinsker wrote:

> The problem with this, other than finding the coil someplace, is
> that the laminations are riveted together, and I guess I'd have
> to drill out those two rivets to get out the coil that is linked to
> the laminations and put in the replacement, and I suppose I'd have
> to somehow replace those rivets.

So drill out the rivets, carefully, and replace them with appropriate
bolts when you re-assemble the thing.  The object is to try to maintain
the insulation between the laminations.  That means that you'll want to
use bolts that are of slightly smaller diameter than the holes in the
laminations.  If there's an insulation sleeve around the rivet, put it
back in around your replacement bolt.  None of this is overly critical,
however: the worst that can happen is that the motor might run a bit
hotter than normal, and there's generally a lot of leeway there anyway.

To rewind the coil, you first need to find someone who can help you
identify and obtain the wire used therein.  It's generally a good idea
to use something close to the original gauge of wire, so borrow a
micrometer and measure the wire diameter.  You'll then need to find
some appropriately-sized magnet wire, and this can be an interesting
search.  I'd guess that your coil might have been wound with something
around #30 AWG enameled magnet wire.  Enameled wire (also known as
Formvar or other names) is needed because the insulation is thin, and
this enables you to wind a lot of wire in a small volume of space.

Then you have to try to get an idea of the number of turns of wire
that were on the old coil.  You can use several strategies, including
unwinding the old wire and keeping count of the turns as it comes off
or multiplying the number of turns across by the number of turns of
thickness.  Neither method is perfectly accurate, but again, this isn't
all that critical as long as you've done a reasonable estimate.  Expect
a big mess of wire as you're unwinding the coil and counting turns.

Don't try to salvage the wire from this or any other motor.  You'll
drive yourself nuts doing so.

The most important point is that you'll need to build a winding machine
of some sort.  This can be quite crude, i.e. a coat-hanger-wire crank
held in screw-eye bearings.  A hand-cranked drill works quite well.
Don't use a drill press or other electric-powered device unless it can
be made to spin very slowly.

Once you've got the winding machine made, you'll have to re-wrap the
insulation around the core.  This can be either Teflon or fiberglass
tape, with the latter being favored.  I haven't tried either of these:
Teflon tape, however, is available at your local plumbing department
for pipe-sealing purposes.  Heat-proof tape of some sort (it doesn't
have to be adhesive) ought to be available in some context or another
amongst the electrical supplies.

Once the insulation is applied around the core, just rewind the coil.
It doesn't have to be particularly even or orderly, though it's nice to
do a job that you don't have to hide beneath more tape.  I generally
wind up doing the latter when I make electromagnets for How Things
Work.  Again, try to get all the turns onto the core, but don't despair
if they won't all fit.

In general, your best efforts will be good enough for a small motor.

If you find a good source of magnet wire, let everyone know about it.
The conventional wisdom sez to ask at a local motor-rewinding shop.
This presumes the existence of a motor-rewinding shop that is
reasonably friendly, and I have never found such a place.  You may
have better luck.  You can also ask the shop to rewind your coil for
you, but that's pretty unpleasant as well.

The upshot is that yes, you can rewind the coil yourself if you can
find the wire to do so, and it's generally not too hard to do so.

Mark Kinsler