Some thing to look at first. When you turn on your key, do your headlights come on? When you press the starter button, do the headlights go out? This will confirm you have power to the relay and the starter button is ok. The starter relay turns off power to the headlights when starting the bike which indicates it is a changeover relay. I think Triumph uses a type B, 5 pin changeover relay (see below). With the key on you should have 12 volts (battery voltage) on pin 30 and pin 87a (power to lights). When pressing the start button you should have battery voltage on pin 85, 86 powering the pull in coil and on pin 30 and pin 87 going to the starter solenoid with 0 volts on 87a. If the relay is clicking and nothing is coming out 87, then the contacts inside the relay could be fried. Pull the plastic cap off the relay and have a look.