Intermittent No-Start / Clicking - Worn Starter Contacts, Bad NSS, and Corroded Battery Cables
1987 Supra NA automatic would intermittently fail to start. Only a clicking sound from the passenger side starter relay was heard with no cranking. The problem was progressive - initially lasting 10 minutes before the car would start, then growing to 30+ minutes, and eventually the car would not start at all until the next day. Jumping the car did not help.
Three overlapping issues: (1) Worn starter solenoid copper contacts - the flat contact surfaces had lost material where the plunger touches, requiring extra plunger travel and causing intermittent connection. (2) Dirty/dried-out neutral safety switch (NSS) contacts - the 35+ year old factory grease had dried and crusted on the copper contacts inside the transmission-mounted NSS, preventing reliable electrical connection in Park position. (3) Corroded and poorly connected battery cable terminals - both positive and negative battery connections had corrosion, loose fit, and damaged wiring including a disconnected fusible link wire.
Moderate confidence — Original poster confirmed this resolved the issue