I suspect that there are must be two different effects: one that causes the car to stop, which would be more or less immediate in effect, and another that dissipates after some minutes.
For the immediate effect that causes the car to stop, this would most likely be electrical, such as an induced surge in the automobile's Engine Control Unit (ECU). According to a NOAA site about lightning safety, "damage to the antenna, electrical system, rear windshield, and tires is common" in automotive lightning strikes.
For the longer term effect that might prevent the car from immediately restarting, the effect is probably either:
- thermal - an obvious effect of a lightning strike is the rise in temperature of objects and atmosphere in the immediate vicinity of the current path of the lightning. If the incoming air into the engine is superheated, the effect would be that the fuel/air mixture would be off far enough to prevent the engine from starting. Other possibilities could include the temporary vaporizing of fuel or electrical heating of the starter motor causing clearances to temporarily change, making it require more energy to turn.
- residual electrical - the vehicle's ECU has memory which stores parameters about the engine. In the event that these parameters are damaged or erased, the computer needs to effectively "re-learn" these parameters (such as ignition timing fine-tuning, and fuel injection timing), possibly restarting from factory defaults if the engine fails to start after multiple attempts. See this in depth article about the details of ECU operation.