Dmaccole is correct. The 7805 voltage regulator will not function at 5 volts.
The suggestion to run 12volts at 40% is also bad. The MR16 uses pulse width modulaion to control brightness. At 40%, you would be getting the full 12 volts 40% of the time and 0 volts the remaining 60% of the time.
There are two ways to make this work. We will dismiss modifying the board and assume you have both 5 & 12 volts available. Connect the 12 volts to the proper terminals onthe board. Connect the ground wire from the 5 volt supply to the ground terminal on the board. Connect +5v to your strobe or other device and connect the ground lead from the strobe to the approriate channel GND terminal on the mr16. The mr16 swiches the ground lead, not the source lead. Do not connect anything to the pos terminal - that is connected directlt to thepositive input terminal.