The solenoid may not be activating because the engine is simply in gear. Most outboards have start-in-gear protection that stops you cranking the motor when it is not in neutral. There is also a fuse that protects the electrical circuit that activates the solenoid. If this fuse is blown the solenoid won't activate and therefore click.
Working through it with no click
With no click at all I take the starter motor, solenoid and neutral switch out and test each one on the bench. In this case the solenoid was fine and it was the neutral switch (at 13:49): it stopped the power ever reaching the solenoid, so it's easy to blame the solenoid by mistake.
The boat's electrical system starts and ends with the battery. It is important to check that the battery is fully charged and in good health before starting to diagnose any electrical problems.
The fuse supplying power to the ignition key is blown
When you turn the ignition key, a small amount of electrical current is sent to the starter solenoid to activate it. The circuit that supplies power to the key switch is fused, so if that fuse is blown you won't hear the solenoid clicking as it activates.
A failed starter solenoid that no longer activates won't engage the starter and has to be replaced. Links here help you find the correct replacement part number for your motor.
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