I didn't know the 7805 would not disconnect when voltages dropped below 7 volts. But it doesn't really matter as long as it doesn't start passing 5 volts until it gets at least 7. Funny thought about a bird flying over. I tried using my hand to simulate a bird and nothing happens. I think the solar cell is providing almost 500mA at peak so clouds, birds, hand passing overhead, etc. don't lower the output enough for the circuit to stop.
There are few ways allow you to do it. I list 2.
1. LTC4365 - Linear Technology
UV, OV and Reverse $3.69
Supply Protection Controller IC (turn off power)
2. ON Semiconductor
Undervoltage Sensing Circuit (reset Arduino)
I list them here as bandages not solutions, but if I were you, I will go pito -Reply #4 direction.