How do you oil the speedo head in a 61 Valiant?

What type of oil do you use?

There's a small brass cup plug on top of the potmetal casting on the back of the speedometer, in line with where the cable screws on. Use Q-tips (dipped in alcohol if necessary) to clean the cup plug, which may or may not have a small hole in it. Either way, use a push pin to either make a hole or clean the existing one, and put a small drop of light utility oil (Tri-Flow, for example).

Yes, this requires removing the speedometer from the car. While you're at it, check for wiggle in the speedo shaft. You will almost certainly find it, which means your speedo needs rebuilt.

And since you'll be unscrewing the speedo cable from the speedo, might as well service it (linked post contains info on where to get speedo rebuilt, too)

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