You should not have to use an op amp or arduino depending on what you are trying to achieve.
In my car, the backlighting of the control panel dims at night so as not to dazzle the occupants.
On other systems the backlight intensity increases at night and turns off during the day.
I assume that you are after the latter behavour.
I have draw up a rough schematic. Ignore the battery voltage. The voltage should be 5 v.
D1 is you backlight LED. (ignore the "QED123" written next to it).
Check your datasheet for the LCD display and see what the maximum current for the backlight is and using ohms law work out the value of R1. Make sure the current is well below maximum to be safe. I have found that you can be conservative and still have good brightness.
Adjust value R2 to make sure that you completely saturate the transistor at maximum darkness to get maximum brightness of the backlight.
The value of R4 will be dependent on the range of the LDR. Try matching the value of R4 to the maximum value of the LDR to start with and work from there.
Try the circuit using cheap LED instead of the LCD until you get the values right then put in the correct value for R1 (using Ohms law ) and attach your LCD.
Others feel free to chip in to add your changes/advice.