We've got the exact same car and had a similar prob. except when
it wouldn't start it really wouldn't, though after first couple
experiences we would just give up right away if it wouldn't start.

This tended to correlate to high humidity/rain.  After sitting
a day in dryer weather it would work.  Other cure was to take
out the coil, put it in the oven at 150-200 degrees for an hour
or so, and clean the mounting surface and screw terminal...

I actually did this about 3 times before I bought a new coil.
New coil was $109 US. on the net.  Earlier I had seen $200, thats
why I postponed.  Actually there was a slightly cheaper coil right
at the same place, but I went for the apparent better quality.

-------From Drivewire.com
For Vehicle: 1998 Volkswagen Jetta III GL 4 CYL
Part: F2010-32950 - BOS Bosch Distributor Cap - (1) @ $10.49
Core: $0.00, Item Subtotal: $10.49

For Vehicle: 1998 Volkswagen Jetta III GL 4 CYL
Part: F2020-24527 - BOS Bosch Distributor Rotor - (1) @ $4.55
Core: $0.00, Item Subtotal: $4.55

For Vehicle: 1998 Volkswagen Jetta III GL 4 CYL
Part: F3004-55219 - BER Beru Ign Coil amp; Control Unit - (1) @ $109.06
Core: $0.00, Item Subtotal: $109.06

The other stuff seemed ready also, car has 80,000 miles.

So you can play with the coil.  Your problem could be something else,
this is all I'm up to speed on.  Coil is a known issue.

Irv

You have a coil that is shorting internally by the humidity (morning
dew, etc).  It will also happen whenever there is a large temperature
drop overnight.  My 96 Cabrio did the same thing.  I bought a new coil
and that was the end of the problem.  Do it and be done with it.
TEMiller

Thanks to all who replied.

My friend decided to take the car to a trusted mechanic, the kind who
likes cars and enjoys discussing a problem with the customer, if the
customer is so inclined. He sprayed the coil with a slight mist of
water, and got a no-start situation. Replaced the coil (including the
ignition control module (or whatever it is called in the VW world)
mounted at the bottom of the coil). He then repeated the mist test. This
time the car started without hesitation. The mechanic kept the car over
night, and did a test drive in the early am the following day. No
problem found.

So all good? Not quite. The car now starts on the first try every
morning, but it still cranks a good 5 - 6 seconds before catching (as
opposed to 3 - 5 good 10 second attempts with the old coil). This PM,
after the car had ben sitting for 10 hours or so, I hooked a timing
light to it, connecting to the high voltage wire between coil and
distributor. Again, my friend had to crank for a good 5 seconds before
the engine started. The timing light did not start strobing until just
about when the engine finally started. Any ideas?

TIA