Here is a picture of the butterfly.
Very nice!
Each wing is 10x7x0.8 and I don't know exactly how much it weights but its all made out of paper so it does not weight much.
Got it.
Probably the easiest solution is to use a tiny servo (or two)...
https://www.google.com/search?q=tiny+servo
For a board, any Arduino compatible will work
#. From this... http://www.pjrc.com/store/teensy.html ...to this... http://arduino.cc/en/Main/ArduinoBoardMega2560
You will need the Servo Library... http://arduino.cc/en/Reference/Servo
And various other supplies...
Something to link the wings to servos (metal rod, plastic rod, string, wire)
Something to hold everything together (wood sheet, metal sheet, plastic sheet, even cardboard may work)
"Hinges" for the wings (metal loop, eyelet, string, paper strip)
Power supply for the board and the servos. If the servos are especially small you may be able to use just one power supply.
# With at least one caveat: I don't know if the Servo Library has been ported to the Due.