1993 Bronco...Turned the key and "click". Brand new fully charged battery, checked the starter, replaced starter relay... now when I turn the key.. not even a click and interior doom lights are dim and I hear a "whining" sound from under back of truck (possibly fuel tank). What does this sound like?

 

The whine is the fuel pump priming and it happens every time you turn the key on. There is a red wire with a 1/4 inch spade terminal on the back of the starter. This spade terminal has no insulation over the terminal itself. I have replaced this on every one of my Broncos simply because with it being under the truck and exposed to road crud the terminal loses its ability to maintain a solid connection. This causes the symptoms you described with no other problems present in the electrical system.

If this turns out NOT to be the issue, check the resistance through the positive cable between the start solenoid (driver's inner fender) and the starter itself. Battery cables may be carrying battery voltage to the starter but it only take one good strand of the wire inside to get the voltage down there. Getting the necessary amperage through this cable requires ALL of the strands. Check all of the positive cables between the battery and the starter for resistance if necessary... it should be very very low (near infinity). You will have to disconnect the cables from these components to test them or you will get resistance readings from the windings in the starter and the solenoid and you don't want that.

These are just two possibilities but two that may or may not be readily apparent after your initial checks.

 

I bet that wiring has a loose ground.