Arduino in car: capacitor for extra 3 sec of power
Don't use the 12V\$\rightarrow\$5V regulator, the Arduino needs at least 7V in. Use the battery's 12V directly instead.  

The capacitor's value will depend on the Arduino's power consumption. The Arduino webpage doesn't say what the Uno consumes, so you can't say right away what capacitor value it needs. In any case it's not designed for low power. I checked the datasheet for the voltage regulator, and that alone already uses 6mA. On the schematic I can see two microntrollers: an ATMega16U2 running at 16MHz, and an AtMega328P, also at 16MHz. The former may consume up to 21mA, the latter says 9mA at 8MHz, so it's safe to say 18mA at 16MHz. We already have 45mA, let's round it to 50mA for the other components.

If a capacitor is discharged at a constant current, then

\$ \Delta V = \dfrac{I \cdot t}{C} \$

You start at 12V, and the Arduino needs a minimum of 7V, so \$\Delta V\$ = 5V, I was 50mA and t = 3s. Then

\$ C = \dfrac{I \cdot t}{\Delta V} = \dfrac{50mA \cdot 3s}{5V} = 30 000\mu F \$

That's the minimum, I would pick a 47 000\$\mu\$F/25V capacitor. Add detection for power off, so that you can switch off all unnecessary outputs which also may consume current, a relay for instance.  

If you want to know exact what the power consumption is, add a 1\$\Omega\$ resistor in series with the power supply and measure the voltage drop. A 50mV drop means 50mA consumption.

Also add a TVS (Transient Voltage Suppressor) at the Arduino's power input; a car's 12V is extremely dirty.

Add the diode clabacchio mentions. A series resistor of 10\$\Omega\$/5W will charge the capacitor in 1.5s when applying power.