Hantronix 20x2 LCD backlight
If it is an EL backlight, then you'll need a "high" voltage AC source, typically around 100V-400V at 50-400Hz. It's a tiny current, so the light is pretty efficient despite the high voltage.

Only trouble is it's a bit more complex to generate the voltage, this link shows how to build a basic boost converter for an EL backlight. If you google for "EL driver" or similar you will find more examples:



In the above circuit, Q1 switches on and current flows through the inductor. Then Q1 switches off, causing a high voltage to appear at the drain of Q1 (charging C1 through the diode). Q2 then switches on to put the voltage across the 100k&Omega; resistor. 2 Timers on an AVR are used in this example. You can adjust brightness by varying the frequency or duty cycle on Q1. Q2 controls your AC frequency.    

(just in case - if it is a standard LED, you would need to apply the +V to the A, and ground to K. If it's a white LED with >3V drop then at least 4V may be needed with a series resistor to limit current. However, looking at the back it seems the jumpers for what may be the LED option are not connected, and as Anindo says it does look like an EL panel poking out under the display)



According to (what I think is) the datasheet, it seems there are various option for the backlight.