Hero999 hit the nail on the head, as he almost always does
Although, I have seen 'consumer' products dimming LED's (for maybe 2 dimming levels) using basic analogue, varying the current....its not really a good way of doing it. I will say this though, providing you have a current limiting resistor in series with the LED backlight, using two different voltage levels WILL change the brightness, but you will have to experiment with what levels work, its not as simple as 'halfing the voltage to half the brightness'!
For just having 3 levels...fully ON, fully OFF, and 'somewhere in between' it would probably be over kill to use a PIC, or fully controllable PWM. I think a good half way point would be a simple 555 timer, or possibly a two level current source.
A 555 timer can be used for PWM, with a standard astable circuit, connect your '0-12v dimming' control line to pin 5 of the 555 via a voltage divider (if the 555 is powered by 5v, don't give it 12v on any pin!).....or even simpler, connect the 'dimming' control line to the 'reset' pin. When the 12v is high, the timer will run at its configured frequency, and duty cycle (ergo: with the correct part values, it'll be 'dim') and when the 12v source is 0V, the 555 will stop running, and leave its out 'high', (ergo: full brightness). If the control line works the other way, then a simple transistor inverter would do.
I'm just spitting out idea's here man, its what I do
If you need anything more, I have some free time at the weekend for a rough schem.
Yet another idea: Although even new cars still insist on using crappy 'bulbs' in the dash for lighting, if there are any LEDs in the dash, you could check those to see if they are dimmed by current, or PWM. If its PWM, then you could just 'tap' that line, use that to turn on/off a transistor with the power to your backlight. (don't do it direct, as you need 3v, and the LED's circuit in the dash may be muich higher).
Well thats food for thought at least....at most I'd say a NE555 timer chip, with a few resistors/caps and possibly a transistor or two........at the very least maybe a transistor and a few resistors....to be honest I'm half asleep so working out specifics ain't on the cards right now
Cheers,
Blueteeth