neepie:
there is a 100ohm resistor on the LCD. That's actually not enough(?)
This will be marked as "R8", and usually an adjacent zero ohm resistor R9 is in the other line to the LED. 100 ohms across 5V would correspond to 50 mA but a white LED will have a voltage drop of no less than 2.5V, meaning the current will be less than 25 mA. An Arduino port should cope with this OK, certainly if you are not loading other ports significantly.
Mind you, I do wonder why people want to PWM the backlight on these LCDs. You only ever actually want two options, either you want to turn it off to save power, so you just turn it off or on, or you think it is too bright at night, in which case you bypass the transistor with a 1k (or so) resistor to give it a minimum level.