Condition Car Ignition Signal

I'm assuming you don't want to use a microcontroller for this, so here's a 555/logic based idea:

Disclaimer: I threw this together in 5 mins, and haven't used a 555 timer for ages, so I may have made a mistake (or three)
Anyway, the basic idea is simple, set the 555 up in monostable "one-shot" mode, and use an inverter gate and a couple of caps/diodes to produce negative pulses on ignition on/off. The inverter gate can be anything that has a reasonably low output impedance.
The inverter power pins are not shown, and ignore R3 (it's just there for SPICE simulation purposes)

Here's simulation of the circuit (on/off time of ignition shortened from "normal" use)
The ignition starts the simulation on, turns off at 1 second, and turns on again at 11 seconds, each time producing a 3 second pulse at the 555 output (this can be used to drive a transistor to switch your GoPro):

R1 is a pull-DOWN intended to keep the gate of the NFET closed.
BAT54 comes in a few flavors; the one in the schematic is the single-diode version (with 3 pins).

Edit (per mkeith): BAT54, really?! This should be a ~15-16V zener.

The "hall sensor" most-likely includes biasing and a transistor/darlington pair that is not shown in the schematic. In any case, a typical FET could withstand 20V, so clamping it slightly above the alternators voltage will keep it inactive but ready, just in case.

Here is the "original" semi-solid-state ignition design; this design has been upgraded, over time, with parts being replaced or added for protection of the components and/or stabilization of it's operating point. Manufacturers re-design and re-implement this circuit and change the names of the signals "every 4 years" but the function remains constant:

^{simulate this circuit – Schematic created using CircuitLab}

R1 is ballast, providing weak ground to coil.
The NPN provides a strong, sudden, ground and the coil reacts.

At their heart, ignition systems are just "boost converters".