[bobski]

If his battery is on its last legs, cranking the starter could be pulling input voltage below the regulator's minimum. That could drag down the voltage to the processor and trip a brownout reset. The MPGuino is set to be more sensitive to that, right?

To answer the original question, yes it's possible to add a backup power source (a 9V might work) as long as you use a pair of appropriately sized diodes to keep the two power sources isolated from each other. Without the diodes, the car would try to charge the 9V, probably destroying it, and the 9V would try (and fail) to power the whole car if the main battery were disconnected.