Drive 12V 1A with 3.3V 10mA (the cheap & easy way)
I thought of a 3V relay (e.g. G5V1-3), but this requires more than
  10mA of power.




Most folk would use a 12 volt relay coil controlled via a BJT like the above. Power is measured in watts or mW; Current is measured in amps or mA etc..

The diode is to protect the transistor when it deactivates the relay and magnetic stored energy is pushed back into the circuit.