The resistor value isn't too critical. About 1K should be fine.
The gate has a small capacitance. That capacitance (with no resistance) is bad for the Arduino when it switches state because the capacitor gets charged/discharged without safely limiting the current.
When you add the resistor you limit the current but you also create a low-pass filter which makes the MOSFET switch on & off more-slowly. A larger resistor makes it slower. During the time it's switching (when it's neither fully-on or fully-off) it dissipates more power and generates more heat so you'd like to switch it fast.
Both of these a bigger issues when switching high-power at high frequencies (such as PWM or high-frequency PWM).