This project has two components, one for the thunder (sound) and the other for lightning. The important part is that they are both synchronized together - the light has to flash the same time that the thunderclap starts.
I had a very hard time finding an inexpensive way to do this by just purchasing an off-the-shelf thunder/lightning strobe light. To get something that looks even close you have to spend several hundred dollars. I just wanted something cheap for a Halloween prop so I went about thinking this project up.
Getting the strobe light to be triggered by the Arduino was the hardest part. Off-the-shelf "lightning strobe lights" are just strobe lights that flash in an interval set by turning a knob to get the desired flash rate. In nature, lightning is much more random so the off-the-shelf stuff didn't really work for me.
I was able to figure out the strobe light circuitry enough so that I could use a relay to close a circuit which would flash the light as long as the circuit was closed. I originally wanted to take the necessary circuitry out of the strobe light and put it in the project but found this way to be much better. I could reuse the strobe light for its original purpose and I could keep the modified strobe light wiring safe to use.
After I figured that out, I needed the Arduino to quickly power up a relay that had enough current capability for the strobe circuitry. There is an abundance of large relays that require 110 volts AC that would have done the trick but I wanted the coil to be controlled by the Arduino, so I found one that only required 5V DC (3 volts actually, Goodsky RW-SH-103D). I didn't have much luck getting the relay to work by driving it directly with the Arduino digital outputs so I wound up using circuitry I used in the darts project - a high-power Darlington "driver" transistor (TIP120) that would trigger with low current and voltage but supply a larger current and voltage.