It would be reasonable and logical that the starter motor or starter solenoid could be the source. Although it doesn't rule out an ignition switch.
Two things in my experience.
If rapid clicks, starter gear probably not fully engaging with flywheel due to insufficient current.
Low battery (may be charging issue if battery is good, Any independent shop will check battery/charging) OR the solenoid is shot (copper connection pads get worn out).
If it is intermittent solenoid, single click but no turn over then try again and works, solenoid is bad as above. Usually this deteriorates over time especially in cold climates like New England.
If you believe solenoid, pull starter out and take to auto electrical shop for a rebuild. If cold climate, the starters take a beating and they will rebuild starter & solenoid.
Folowing not likely but...late models use a relay. The ignition switch controls the relay, the relay feeds the solenoid, solenoid feeds starter. I did have a 1997 Tahoe that was intermittent (turn key twice it starts). I did not believe solenoid was bad since I'm warm climate. This can be (carefully, know what your doing & verify relay part numbers) checked by substituting a relay from another "non-vital" circuit (say air conditioning). The relays MUST be the same so verify part number.
My Thaoe's electrical diagram did NOT show the relay and I almost rebuilt starter. I could not believe solenoid was issue. It was bad relay, $23 at NAPA.
I would be suspicious of a weak battery first because it is only during cold weather. This is when the battery is most susceptible to failure. Drop it off at your local independent shop that you use outside the night before when it is cold, and with a little luck it will happen so it can be diagnosed.
This is caused by not enough power reaching the starter solenoid to make it engage the starter motor. The culprit could be: a) a weak battery (only if it turns the starter slower than normal when it does turn it)b) worn ignition switch contacts c) loose wire connection (somewhere from switch to solenoid) d) a wire that has developed high resistance or become frayed or worn out. e) worn starter solenoid f) on newer models an ignition relay
I had this problem on two older model cars I owned, on both my cars the cause was the ignition to starter solenoid wire had developed high resistance. This is the reason newer models include a relay. I adapated a relay on both my cars long before this addition was included on newer models. This I�ve found is the main cause for clicking starter solenoids, and is very common on earlier models but many times missdiagnosed as starter, solenoid or ignition switch related.
It could also be a bad battery connection. Check the battery cables for tightness and sometimes corrosion.
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