What are the causes and solutions to battery-powered LCD backlight buzz/hum?
That kind of backlight is an electroluminescent panel. This type of panel requires a fairly high AC voltage to activate it, provided by an inverter.

Such inverters can generate acoustic noise through a variety of mechansims, including magnetostriction in the transformer coil, to piezoelectric effects in capacitors, to transformer windings that are physically loose. The first step would be to determine what the source of the noise is.

Sometimes, what would otherwise be an unnoticeable noise gets amplified to objectionable levels by a resonance in the product's case. This can be addressed by either eliminating the resonance, or decoupling the noise source from the case, typically by incorporating some sort of elastomer (e.g., rubber) into the PCB mounts.