I drew up a Schematic for the typical HV Pulse Ignitor, but must have never posted it on line!
It consists of an array of Voltage Doubling Diode Matrices, set up in multiple tank circuit formations (RC Charging Circuits).
The pulses typically peak out around 2500 Volts, with a quick duration of maybe 10 ms. Current is in a fat level (possibly as high as 500ma - upto 1500ma), so the complete pulse is a short high Joule event.
Pulse is only required to jump current between the Cathodes within the arc tube. Pulse Amplitude (Voltage and Current) is a result of the Impedance which is connected to it - higher Z results in higher Amplitudes. Once the arc tube has an established plasma, the lamp becomes very low Z (actually negative R), so the ignitor produces very tiny pulses.
Input to the ignitor is regulated by the Ballast (of course), so the ignitor (may) need some method of current Ballasting to prevent overcurrent draw (not sure if this would apply to bench testing, but make sure first!).
To test one, either perform the routines given by Rick's post (best method overall! Thanks, Rick!), or bench test using a High Voltage type meter with a regulated current output power supply.
You will need to view the data on a scope, unless you have a HV sample and hold feature for current and voltage.
Look for pulse width, Amplitude and frequency. Pulses should be almost on top of each other - close to 100 Hz or higher. Current and voltage amplitudes should be near constant levels for at least 5 minutes. Excessive harmonic distortion would indicate too much resonance within the tank matrix, leaning towards failing ignitor.
Harmonic distortion will (should) be even overall, and result in the pulses having jagged steps with wavy edges (???).
Residual stuff from charge and discharge time constants will cause sequential steps in the pulse, and internal resonance will distort the complete pulse envelope.
Anyhow, if you are able to see high peaks (>2KV and >500ma) consistantly, the ignitor is useable. Need to push these pulses into something with an extremely high Z - like a CRT or a brand new HPS lamp.
p.s. the most effective and productive HV pulse ignitor tests are as Rick (Rmeill) explained. Try swapping with known operable ignitor across known operable lamp. Test Ballast OCV using volt meter or "Wing It" with alternate lamp types (Incandescent or H33).
Contact me here or directly if you have further / indepth questions.