Robin2:
That suggests that the input wire is normally at 12v (but car voltages can rise to 15v) so you would need a voltage divider to bring that down to a max of 5v for the Arduino.
As it's automotive I'd prefer an optocoupler there.
But, it's a switch, not a normal signal:
What about the controls that are currently connected to this button? Those controls are to be taken over by the Arduino as I understand it, which probably means that the button can be completely disconnected from the existing controls, and instead be connected as normal button to the Arduino. So the voltage now at that button becomes irrelevant. Important is that it's really a momentary button (a multimeter can confirm this in an instant).
The lights themselves are connected to relays, which can be controlled easily by an Arduino. The relay operates at 12V which is fine, you anyway need a transistor to switch (you probably need a high side switch, in automotive the grounds are usually all connected to the chassis and the positive lead is switched - so PNP or p-channel MOSFET). Here I'd definitely want to add an optocoupler to isolate the Arduino from the vehicle power system.
Then the power supply itself: get an off-the-shelf car phone charger. Those are designed for automotive, so have proper protection circuits, and produce 5V which can be power the Arduino.