Arduino spuriously resets with power draw
With the schematic I see you have diodes in both power and ground.
The one in the ground side is not necessary and is probably complicating how you measure any disturbances because your scope will also be connected to ground. Also it would be better to use a Shottky diode as it has less voltage drop.
It would be useful to have a scope trace showing both the motor power and the Arduino power simultaneously. expand the trace so we can see the width of the pulses to determine the likely cause. (it is better if possible to do a screen shot using the scope rather than taking a photo, it will be clearer).
If the width of the dropout is consistent with the time constant of the large capacitor and the drain of the processor then it is being caused by a power drop-out, if it is very short (nanoseconds to microseconds) then it is likely interference in some other way.
It may be worthwhile setting the brown-out-detector(BOD) voltage to lower than normal (say 2.7V) to give more margin if the voltage does drop.
A number of years ago I designed a similar system. In order to avoid the voltage drops caused by the heavy starting motor load I stepped up the battery voltage to 6V with an LT1949 boost converter, stored energy in a large 2200uF cap) then regulated it down to the 5V needed by the processor (ATMega128). The extra 1V stored in the capacitor allowed it to drop by up to one volt without affecting the supply to the processor, and another 500mV or so before getting to the minimum voltage requirement for the processor. This is three times the margin you get by just putting a large capacitor on the supply.
That system allowed the battery voltage to drop down to about 3V without trouble and could survive 10s of milliseconds with the voltage even lower.