You can do that, possibly.  It depends on whether or not the display has an easily replaceable CCFL bulb.

What you'd need would be an LED strip, the appropriate resistors and LEDs.  You'd need a variable PWM source or perhaps an additional resistor/variable voltage regulator to give you the brightness you want, or just adjust the resistors (which can get messy if you do it too many times).  You can adjust the resistors to work profitably on a 5 volt supply, for instance.

Another option is to replace the inverter with a variable brightness one, and put a small pot in to control that brightness.  If the existing configuration has the provisions for bright/dim the inverter, you may want to mess with that instead of putting LEDS in or replacing the inverter.

I have replaced CCFL lamps with LED strips, and do have LEDS and the blank PC boards, though.  You'll want to be able to do surface mount soldering, the parts are 1206 and 805 in size.

I'd look at messing with the control of the inverter first.  I would NOT mess with the high voltage part at all, you're talking 1600 volts at 8 ma or so to light that little CCFL lamp.

Likely the first thing would be to see what the inverter has in the way of a control voltage, for the ones I've seen, it's a simple voltage from, say, 0 to 4 volts.  Don't remember if more is less here, but that depends on the inverter.  Some are not brightness controlled, some are.