Simplest way to turn on a LED controlled from a 0v control signal?
This is one of the simplest ways, assuming the input voltage is equal to (or a bit higher than) the voltage the LED operates from.
You could use a P-channel MOSFET and eliminate R1, which is slightly simpler, or use a single-gate logic chip which would be the same number of parts.
The input should be pulled to ground (not just left open) for the LED to illuminate.



simulate this circuit &ndash; Schematic created using CircuitLab
But the simplest way is simply to connect the LED + resistor between 5V and the input, if the input can drive it acceptably. That's only two parts. Or only one if you use a LED with built-in resistor.