Electronic – BJT vs Mosfet, switching variable current battery powered device

Since you will be using transistor as a switch the rule of thumb for a saturated BJT is to use beta (current gain) of 10 or 20. Meaning you can directly drive a general purpose BJT (with at least 100mA of collector current) from your Arduino pin.

This is how I would do it:

^{simulate this circuit – Schematic created using CircuitLab}

NPN always has a base voltage ~0.7V higher than its emmiter. Since the emmiter here is grounded the base voltage sits at around 4.3V (5V that the Arduino can provide -0.7V). And a base current of 10mA gives us 4.3/0.01 =430 Ohm, but you can use a more standard value of 470 Ohm.

Diode D1 is there to dissipate the energy stored in the relay coil when the transistors shuts off. You can read more about this here on EE stack exchange.

There are two issues with your design. First and most important, replace that 100 Ohm resistor which is way too small with 10K. Second, the Vgs On of the mosfet could be bigger than the battery voltage. You may want to use another battery or another FET