Ok I did tests with various resistors and with fresh batteries. I got optimal results (brightbnes/battery life) using the 5v line and a 300ohm resistor. The light runs at 2.3v and is bright enough. With a fresh pair of alkaline Duracell (mn2400) batteries I get about 6 hours and 30 minutes of play time. With rechargable Fujicell batteries (1100mha fhr 4uex) I get about the same play time. Note that the rechargeable ones run at 1.2v (I noticed that just now).
Judging by the voltage reading I get from the batteries after the GameBoy turns off it seems that they are pretty close to empty so I will stick with this setup.

@lw I think testing with various resistors would be the best solution. For example you can start with a 200ohm resistor and depending on the results go up or down.

I had no idea about this, it sound very possible considering the behavior I am experiencing.
Today I installed a 300ohm resistor and the heat was reduced again, additionally I was able to power the gameboy for about another hour and a half using the same pair of duracell batteries. They are still far from fully drained after the gameboy powered down.

Currently the backlight is connected to the 5v line on the cartridge slot and it runs on 2.3v with the resistor, but I saw there is also 3v line around the voltage regulator, so tomorrow Im going to try connecting to that one and see if there is further improvement.

Using a LIPol battery is an interesting alternative, but I kinda like being able to swap the batteries, so for now Im gonna stay away from this.

Thanks for the input.

The voltage board still heats up, it gets less hot, but still hotter than normal. Also I got a pair of Duracell turbomax batteries that can show you how much charge is left. The after about 2 hours the gameboy powered down, yet the batteries themselves show 100% charge which is very strange. 2 hours of battery life is pretty abysmal, I expected some reduction but this is way too much and Im seeing people claiming they get about 7-8 hrs...

I guess I will buying various resistors and testing them out today or tomorrow and update with the results. Meanwhile if someone has any ideas Im all ears (or rather eyes).

Hi hi smile
I just modded my GB Pocket with a backlight and the screen looks quite good. At first I did not use a separate resistor as the backlight should have internal one, however I noticed that while playing the lower left corner of the gameboy would heat up a bit too much for my taste. So I added a 39 ohm 0.25w resistor and it heats up less now. However I also noticed that my batteries only lasted about 2 hours (with the resistor added) of playing Pokemon Gold (mid contrast, max volume). It is true that I have charged those batteries about a week ago, but I have used them very slightly before, so they should have lasted more than 2 hours, way more - those are 1100mha batteries >_<.

Have any of you experienced heavily reduced battery life after installing a backlight? Would adding a stronger resistor (say 100ohm) fix the heat and battery life?

I dont really know the model of the backlight, it appears to be made by ASM and has 4 LED`s (for reference it is from gameboymodsuk).

Any suggestions are welcome.