If the clicking is coming from the fuse box, it's likely to be the starter motor relay. I can't see how a relay could fail in a way that will result in it switching on and off rapidly, so that makes me think there is a low battery voltage or high earth resistance causing it to act in such a way.
Can you measure battery voltage across the battery terminals when your trying to crank the engine? You may see a static battery voltage of around 12v, but when you apply the electrical load of the starter motor to it, the voltage could drop off significantly. This will have an adverse affect on all electrical consumers and components on the car.
A scenario to possibly explain why the starter motor relay would rapidly switch on and off: Battery voltage is at 12v, ignition switch is switched to the start position, the electro magnet in the starter relay is energised and pulls the switching contacts together, the starter motor solenoid is then energised and switches power to the starter motor, electrical load of the starter motor pulls battery voltage down to less than 6v, the drop in battery voltage causes the electro magnet in the starter relay to collapse and the switching contacts open, the starter motor solenoid is powered down and the starter motor disengaged, with the electrical load of the starter motor relieved from the battery, battery voltage returns to 12v and the process starts all over again.
If this is the case, then that leaves a few questions. Is there a poor live connection to the battery? Is there a poor earth connection to the battery? Is the battery in good condition (even though it is new)? Is the engine locked and the starter motor is not physically able to crank it over?
That last question makes me ask you, are you 110% sure that no foreign object has been able to drop down into the engine through the throttle body?
If you are unsure on this, I would suggest removing the spark plugs, lifting one front wheel off the ground (both if you have a LSD), putting the gearbox in 4th gear and then rotate the wheel to see if the engine turns over.
If it does turn then you can start to look at possible electrical faults, if the engine will not turn, you have a problem.