Shotgun aproach to repair: Just replace the suspect components
I checked out the bulb and indeed one of the leads was not soldered well to the base. I re-soldered but still I have intermittent operation.
Next I soldered jumper leads right from the leads on the back of the socket to the bulb and still intermittent operation (the ignition voltage always makes the tube glow a little, but no full discharge).
Some strobe web sites indicate an old tube may be hard to fire. New tubes can be purchased from B&H but they are expensisve $100 to $300.
This pdf page from mouser electronics has a nice replacement strobe tube and a new ignition coil/trigger transformer for just a few dollars apiece: http://www.mouser.com/catalog/629/127.pdf
Note flashbulb 'C' and trigger transformer 'A' these both look similar to what I have now and would probably make good replacements.
One last thing to try before replacing the tube and or the ignitor coil is to try to put some extra wire around the tube to see if it will ignite better....