Controlling the power from a 9V battery with a potentiometer
First of all, is 9V the supply voltage of the backlight? LEDs usually have voltages of about 3V, and a higher voltage usually destroys them immediately. Maybe, there are multiple LED in series and a resistor in your display.

Your problem is the characteristic U-I curve of LEDs. A little lower voltage, and the current drops significantly, and the LED becomes much dimmer.

If your potentiometer is too large, a small angle will already cause a high resistance, which does not allow enough current to flow through the LEDs. Hence, a smaller potentiometer may help. Also, have a look at logarithmic potentiometers. As said, the correlation of angle and resistance is logarithmic, so the resistance will change only a little over a wide range of the angle. This allows a very fine adjustment of the current.

However, I wonder that the LED is fully off as soon as you touch the potentiometer... Also, you have to take into account the power dissipation of your backlight. Potentiometers usually can not dissipate much power, and depending on your backlight, you may overload it.

Another solution would be a PWM, for example using a NE555. Here is an example from

http://www.reuk.co.uk/LED-Dimmer-Circuit.htm :



The benefit is that there is a linear potentiometer, and the output power also depends linearly on the potentiometer setting. Also, the behavior is independent from the load connected.