Driving the solenoid can be done several ways.http://www.arduino.cc/playground/Learning/SolenoidTutorial
In this case a 12 volt power supply was used to switch the solenoid, but it could be transformed to operate on 24 volt. The circuit has an optocoupler to protect the arduino, which isn't really needed by the way.
A relay could also be used.http://arduino.cc/playground/uploads/Main/relays.pdf
Next, you'll have to keep track of time.
The arduino doesn't have an RTC on board, so you'll have to add one for a few bucks.
You might... also use a light sensor to determine whether it's day or night, but that may cause little mist in winter and perhaps too much in summers.
It is possible to turn on the mister for a determined time several times a day, but you'll have to write the program for it.
Since you're more or less "playing God" and have an living animal depending on your skills, it's best to be sure it works perfectly. I don't know much about chameleons, but know of a few DIY-projects with fish that decided to swim belly up after automating fish-tanks while having a few programming errors.
If you would like to you could also integrate other stuff like a humidity-sensor, temperature sensor.
That way you might be able to check how humid the air already is and adjust the misting automatically.
A lot is possible I guess.