Making Solenoid valve work with Arduino and TIP120 transistor
The voltage drop you are seeing under load is normal for this type of transistor. Internally the TIP120 is actually two transistors, with the Emitter of the first one connected to Base of the second one. In this configuration the total current gain is very high, but the saturation voltage is also quite high (up to 2V at 3A), because it is the sum of the first transistor's Collector-Emitter and second transistor's Base-Emitter junctions.     



In many circuits this higher voltage drop isn't a problem. However your solenoid valve is only specified to work on 12V +-10%, which means it will just work (or not!) at 10.8V. A device will often still work outside of its specifications, but apparently yours doesn't. Therefore you have 3 options:-


Use a higher voltage power supply (eg. 13.2V) to compensate for the voltage drop in the Darlington transistor.     
Replace the TIP120 with a device that has lower voltage drop, eg. a 'Logic Level' power MOSFET. 
Use a valve that is specified to work at 10V or less. 


Note: Solenoid coils have very high inductance. When switching off this generates a high voltage spike as the magnetic field collapses, which can kill your transistor. To prevent this you should wire a diode (eg 1N4001) across the solenoid, with Cathode to the positive power supply connection so it doesn't normally conduct. Don't get the polarity wrong or the diode will short circuit the solenoid!