I want to know how much current a 16x2 (1602) LCD draws when the backlight is OFF and ON?
NB: My multimeter is broken.
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Typically the LCD driver draws around 1mA or 1.1mA, not counting the external contrast voltage divider. They are fairly consistent between suppliers, as they are mostly based on an (originally) Hitachi-designed chip set HD44780/HD44100.
An LED backlight will draw MUCH more current and you'll have to refer to the individual datasheet to get that number. Typically in the range 50-200mA. For example, the Newhaven NHD-0216BZ-FL-YBW has typical backlight current of 120mA with a range of 90 to 130mA with a 5.0V supply. It does vary quite a bit from product to product though.