Brand new starter solenoid clicking?

12.4 volts with the car off is rather meaningless. 

What is the voltage at the battery posts with the ignition switch in the START position?? 

Another question.  What is the voltage at the battery posts with the engine idling and the headlights on?  I realize you can only do this when you get your car started.  But let me know the answer to my first question. 

You don't have to be left up in the air when you have access to a voltmeter. 

When I get an answer to at least my first question, I may be able to guide you to check some other voltages. 

What I want to do is find whether the battery doesn't have the energy to power the starter, OR, whether there is contact resistance (corrosion) somewhere in the starter primary circuit (battery, battery terminals, battery cables, starter). 

The voltmeter is the right tool to use, but you need to understand what to do with it. 

-Engine overhaul mechanic and general automotive mechanic since 1972

If the battery is five years old it is well past it's best. The voltage figure does not tell you what amps are available to crank the engine. My bet is your battery is knackered.

that clicking sound when trying to start the car is always the starter solenoid trying to engage ..yes its possible a loose connection but 99% of time its a flat battery 

A smart person would have had the battery and charging system tested for free at an auto parts store before wasting their money on a starter that wasn't needed. You have a bad battery and maybe a bad alternator. All the battery cables should be cleaned on the cable clamps where they contact the battery posts and the negative cable where it bolts to the motor or tranny.

Battery is almost 5 years old. I'd have it tested before you go blow more money on unneeded parts.

Dirty/loose/corroded battery terminal connections? Including ground wire to chassis and engine, not just at the battery. Battery is 5 years old and could be bad, what is the voltage while trying to crank? No load voltage is meaningless. Bad cables or improper use can cause a jump start failure. 

Starter is not getting enough juice or there is a bad ground.  Get the battery charged and load tested or just replace it.  Clean the battery cable terminals and battery posts.  With a good battery, try to start.  Car starts, check the charging system (alternator).

No luck, get a jumper cable and connect one end to battery negative post and the other end to a metallic part of the engine.  Try to start.  Car starts, check for a bad engine ground connection or bad cable from battery negative to engine.

No start, clean the B+ terminal at the starter solenoid (the terminal where the cable from the battery connects to) and clean the cable terminal and hardware that connects to it.  Reconnect everything after cleaning.  Try to start.

No start, get a multimeter, set to DC voltage (under 20V) and connect black lead to the B+ terminal of the starter solenoid  and red lead to battery positive and read the voltage while cranking (key on Start).  If voltage is over 0.3V, check for a bad fusible link (if there is one on the cable) and check the cable from battery positive to the B+ terminal of the starter.  Good luck.