I am looking to rent a manual transmission car in the Seattle, WA
area.  I have looked at the all the major car rental agencies to no
avail.

Try an outfit like Rent-a-Heap-Cheap, if such still exist.

Consider buying an old VW beetle from an individual, use it as long as
needed, then sell it for not much less than you paid.  Your real cost
is the insurance, which will only be the liability portions, not
comprehensive or collision coverages. Cancel the policy when you are
done.

Curious:  are you wanting to teach your kid to drive a stick-shift. If
so, I applaud you.

Most of the regular car rental places do not offer standard
transmissions in the US. However, there are some specialty (exotic)
rental places in major cities that offer high end vehicles with
standard transmissions for rent. One such location in the Seattle area
is:

http://www.bnm.com/parkplace.htm

I didn't go as far as to see the quotes, but they do have a
Mini-Cooper, with a six-speed, which shouldn't be too terribly
expensive to rent.

The other vehicles they have that, may also be standard, are:

Mustang
Audi TT
Porsche 
Ferrari
Viper
Corvette


HTH