1969 1 click no crank issue

[XOFD]

Hi All,
I hope someone can help me with this issue.
I bought this car which has been sitting for 7-8 years. Was starting and running ok . I drove it one day and when I went to start it afterwards it just does 1 click and doesn't crank.
As it is for my son I already replaced the solenoid, battery, alternator, cables, ignition switch. tune up including coil prior to this issue. I did notice that there is some type of relay next to the brake booster that is attached to a foam block on the firewall. It will click and then the solenoid clicks after. Does anyone happen top know what that relay is for? Also any ideas on what would cause a drop in electricity to the solenoid. The starter will crank if I jump it directly to the battery with a remote starter switch and bypass the cars wiring.
I'm sure it is a simple wire or connection issue but I am not savvy with electrical.
Thanks very much

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[OneAmongOthers]

Are you sure a previous owner hasn't installed a car alarm at some point? You need to get a volt meter on it ands start checking if you are getting power to the starter or solenoid when you put the key into the start position. start there. If the starter is cranking when power is directly applied it seems like and ignition switch problem will the car start and run if you leave the ignition in the on position after jumping the starter?

 

[XOFD]

No alarm for sure.
Sometimes it has 1 click to the fender well solenoid and when i wire a remote starter switch to the solenoid on the starter it also has a faint click but if I wire the remote starter switch directly to the solenoid on the battery and bypass the wiring harness the starter cranks with no problem.
I think it is in the car's wiring . Just weird how it was fine to start one minute and the next time I went to start it then it would just click
Also I have new ground wires from engine to frame, engines to firewall and new ground battery cable. Still just clicks

 

[ak7an]

I would replace the new solenoid that you just installed with another one. Just because it is new doesn't mean it can't fail quickly. Probably made in China JUNK..