How do I wire up a 2.3-3 V LED to a 12 V car battery that can have a variable voltage depending if the engine is running or not?
LEDs are very tolerant of variation in current.  I would choose a resistor value that will limit the LED current to 10 mA or less with a 14 Volt supply.  The LED should still produce sufficient light when the voltage drops to 12 V.  Many LEDs will produce reasonable light with only 1 - 2 mA.

If you put an LED and resistor in parallel with a fuse or switch, it will light when the fuse blows or the switch is open, but will only allow a few mA to flow.

Edit...







simulate this circuit &ndash; Schematic created using CircuitLab

D1 is the "Power On" LED, and will light whenever power is applied to the motor.

D2 is the "Blown Fuse" indicator.  It will light if the fuse is blown and the switch is on