The number of approach attempts you are allowed will be defined in that Company's Operations Manual (which is approved by the authority).
With the airlines I have operated for you are only allowed a third attempt if there has been a significant improvement in the weather (this also is usually defined). There are also sometimes stipulations that the previous 2 attempts must be on autopilot. (Once again though this would depend on that Company's OM).
So this decision rests with the aircraft Commander (he obviously should be obeying the rules!).
Only exception would be if you have to declare an emergency for some reason.
Another reason for doing approaches onto a different runway is that this might save time and therefore fuel but usually it's because of better RVR etc.