I bought the light (it was 50% off) and took it apart.  It took a while but I got the guts of the light removed.  Inside I found an interesting pulse motor which drove the rotating reflector.  It looked like the motor used some of the same gears as a quartz clock mechanism, powered by a single AA 1.5v alkaline cell.  I tested the motor and found that it ran fine at 3 volts.  The original circuit used a single 1.2v NiMH cell and a small DC to DC converter to drive the small LED.

I removed the small solar panel and glued in its place a 10 cell 5v 160ma solar panel.  I unsoldered the small LED and in its place soldered a much more powerful LED, with the center facing outward.  This would align the LED light with the center of the reflector.  The reflector would produce two flashes of light for each rotation.

I left the pulse motor alone but added a simple LED driver circuit shown below.  I installed a battery pack made from three 1.2v AA NiMH cells.  A 1N5817 schottky diode between the battery and the solar panel prevented current from leaking back into the solar panel from the battery at night.  I used a single Schmitt trigger inverter to monitor the voltage across the solar panel and switch on an n-channel FET at night to turn on the LED and the reflector motor.  I have a short video of the completed lighthouse in action below.