If you are using one of the 3rd party LCDs with a backlight, you need to be a little careful about the current draw of your backlight. The amount of current needed for the backlight varies widely from unit to unit, and the listings on ebay are frequently not complete, or in error regarding backlight current.
Current draw seems to vary widely. I have a Blue 16x2 that is very bright but only draws 16mA. But some of the Green Yellow 16x2 displays that need 150mA to be very bright. In general the blue displays with white backlights seem to draw much less current that others.
You can drive the backlight from the circuitry on the counter PCB. R3 is the backlight curent resistor, but some of the modules have current limiting resistors on the module, others do not. Do not attempt to drive any backlight other than a LED backlight. You cannot drive EL backlights this way.
The backlight LED terminals ae usually on pins 15 and 16 of the connector. Pin 16 is usually the cathode, which is connected to ground. Pin 15 is the anode which is connected to +5v through R3. The LEDs used usually have a voltage drop in the 3 to 4v range, thus a series resistor is needed to drive it from a 5v source. All of this current comes from the voltage regulator (U1). If your kit has a 78L05 (TO92 package) you can only supply about 20mA of backlight current. Using a 100ohm resistor for R3, the backlight current will be limited to 20mA. ((5.0v-3.0v)/100=.02) If your LED backlight is not sufficiently bright you should measure the voltage across R3, and if it is less than 1V, then your LED probably is a 4v type, so you can try a 47 ohm resistor instead. Check to make sure U1 is not overheating.
If this is not sufficiently bright, you will have to change to voltage regulator to a 7805 (TO220 package) Start with a 47ohm resistor in R3. If you need more brightness, you can try 22ohms, or 10ohms. But you will probably need to add a heatsink to U1 because it will get hot. A 7805 in a TO-220 package can supply up to 1 amp, but it will need a huge heatsink for that. With no heatsink 50 to 100mA is the limit.
You can also drive your backlight directly from your 12v power supply rather than the 5v power on the counter PCB. A series resistor of 100ohms will give you 100mA of backlight current. But that 100 ohm resistor would be dissipating 1Watt of power, so you will want to use a properly rated resistor, wirewound 2-5 Watt would be good.