The I2C LCD backpack's I use have a little plastic "shunt" jumper on them that controls the backlight. If you remove the jumper the backlight turns off. It would be easy enough to hook this to a toggle switch. I don't know if they support turning the backlight off with software or not.
Nearly all the i2c backpacks have the jumper. The Jumper functionality can vary between designs but the most common is that the jumper disables the backlight as off if removed and if installed it enables s/w control of the backlight.
But some designs use the jumper to force the backlight to always be on and if removed, backlight can be controlled by s/w.
So it depends on which backpack design you have.
For cool factor, you could use a RCWL-0516 Microwave Radar Sensor module (which are less than $1 off ebay) to detect motion.
If the Arduino doesn't see any "movement" signal from the 516 after a set timeout, it can turn off the backlight.
Similar to a PIR, as Paul mentioned, but uses microwaves.