OK the 5 seconds is arbitrary. Last I checked, a couple of years ago the relays were still available, but if I recall they were around $150. Inside the relay box is a circuit that times out until you get ignition pulses and I'm pretty sure it relies on what is known as a RC time constant which is a property of resistance and capacitance, and with time they can change. If the Fleabay relay is cheap enough it may be worth your while to take a gamble on it as opposed to the cost and availability of a new one.
As far as the socket goes, contact cleaner is about all you can do for it besides cleaning the relay contacts. You could try ever so slightly twisting the contacts to improve connectivity but only by a couple of degrees.
When is the last time you changed the fuel filter??? Also you should not hear clicking from the relay itself, only the fuel pump. The relay will only click once when you turn the key on, and one time more when spark is detected. Once energized it stays energized until you turn the key off...