I love it when people take things apart and expect you to put them back together again...
Firstly, the obligatory warning: Strobe lamps are very dangerous - that entire circuit is live at mains potential, and will (when it's working) be producing spikes of several kV. But I'm sure you allready knew that!
In case you're not familiar with them, strobe tubes are a xeneon discharge lamp with two main terminals and a trigger. The main terminals are at oposite ends of the tube inside the gas, while the trigger is usually a fine wire running over the outside of the tube, electrically isolated from the gas. A voltage is applied between the main terminals which, in itself, is not high enough to ionise the gas and strike an arc. To fire the tube, a very high voltage and very short pulse is applied to the trigger - this couples capacatively through the glass and ionises the gas. Once ionised, the gas conducts a current between the main terminals (lighting up as it does so). The arc will be sustained until the current falls below the holding current of the tube.
I've had a look over your photos, and think it's something as follows:
The tube is directly in series with the mains, via the large diodes D2, D3. This is common; once fired, the tube can conduct for (at most) one half-cycle. On the second half-cycle the diodes block and the tube has time to extinguish. C4 is accross the tube (and seems to be charged through C3 and a diode). When the tube fires, it will absorb all the energy from C4 and then contuniue to conduct for the rest of the half-cycle.
The trigger pulse is produced at the secondary of T1 and applied between Lamp2 and the Trigger terminal. To generate this pulse, Q1 conducts and discharges C1 through the primary of T1 into Neutral. C1 is then re-charged via D1 and R1 from the Line conductor. Q1 looks like it might be a Triac and is driven by Q2 via a capacitive coupling C5. Could this be a relaxation oscilator? I'm not sure if Q2 is driven from the DIP16 or the optocoupler... It's hard to be know how the control is done.
The low voltage supply for the IC seems to be dropped through R2 and the diode that's hiding underneath it, with C2 being the reservior capacator - could that be a zener diode lurking behind it?
Please understand that I can't be sure about any of this, so please satisfy yourself that it makes sense before connecting up mains to your board and ruining it because I was wrong!
I'd start fault finding by checking T1 for continuity; you'll see it only has three connections - the two widings will be commoned at the Neutral point.
Then apply some power (need not be full mains) and see if you're getting any voltage on C2.
It would help us to know the part numbers are on the components, and how the controls and connectors are labeled.
Hope this helps - let us know how you get on (and be CAREFUL).