Changing the brightness on a Hitachi HD44780 LCD screen
Place a transistor in series with a 33R resistor between the LCD pin 16 (backlight ground) and GND and switch the transistor on and off with a digital pin. That way you'll be able to turn the backlight on and off.

You can have a finer control over brightness if you put a PWM pulse on said pin. Just call analogWrite(D3, value) where value is the desired brightness.

The schematics is below. It's from Freetronics 16x2 LCD SHield.