That clicking is the plastic teeth of the gears being ground off because they got out of alignment. It has to come out. Why it happened is to be found out at the car. Read the ASCD thread. Its got the speedometer in there.

 

The problem you will have is getting to the speedometer. You will have to remove the cluster. I find it easier to first remove the steering wheel. There are 2 screws on top of the cluster trim that screw in upside down. Then there are two screws on the bottom that if I recall, you have to remove the steering column trim. On my model, there were 2 screws that held the cluster in from the backside of the cluster panel yours being a 79 might be different. I removed mine when I removed the dash..

If you just want to get to the cable, it twists off at the cluster and then again back at the firewall just above the gas pedal.

 

Adding to palladin's comments... You pretty much have to remove the steering wheel and the two trim pieces that cover the switches on the steering wheel column in order to remove the combination gauge (containing the speedometer). If your '79's like mine, it's actually quite easy (I've done it so many times, I've committed the process to memory):
1) Disconnect battery ground cable
2) Remove the two phillips head screws and lower dash cover on the driver's side and disconnect the two footwell lamp wires and two vent vacuum hoses (they outta be different lengths, so you should be able to pull them off without getting them confused).
3) Pull off the rubber horn pad from the steering wheel.
4) Using a 22mm socket, remove the nut and locking washer in the center of the steering wheel and remove the steering wheel (in theory, you should use a steering wheel puller, but I've only ever needed to give it a few moderate (not hard!) bumps with the heels of the palms of my hands to loosen it for removal).
5) Pull out the metal surround that fits around the key slot.
6) Remove the five or six (the body shop lost my originals, so I'm not positive of the exact number/type of screws) on the bottom of the lower plastic trim piece that covers the steering wheel stalks (turn signal, wiper, etc.) and lift off both covers--one above and one below the steering column.
7) Loosen the phillips head screw at the lower right side of the white nylon combination switch housing so you can slide the switches off the steering wheel column.
8) Remove two phillips head screws in the top of the combination gauge trim and two (larger) phillips head screws at the bottom--one on each side of the steering column.
9) The speedometer cable is actually two separate cables, which screw together on the interior side of the firewall. You'll probably have to unscrew/separate them before you'll be able to slide the combination gauge cluster out. Just make note of how the short section of the speedometer cable routes through the back of the dash as it can be confusing to re-route if you didn't see how it came out.
10) In addition to the speedometer cable, which unscrews from the rear of the speedometer cover, the combination gauge has two other wire harnesses that you'll need to disconnect before you'll be able to slide the combination gauge completely out. Both just press onto pins on the rear of the combination gauge, but since you'll only have a couple of inches of clearance it takes a bit of finagling to reattach them.

This is all covered in the service manual and I agree with heroe's diagnosis. It sounds like something's out of alignment. On the other hand, if the speedometer cable's are dry, that can cause the speedo needle to jump around. That being the case, just slide out the internal wire from the speedometer cable, grease it and reassemble.
Good luck.

 

To add to Freddy's post.

When attempting to slide the cluster out of the dash, you may find it is hung up on something. On the topside, out of sight, are two tabs that prevent the unit from just sliding out. The best way to deal with these is to use two butter knives or two putty knives. You slide each one on either side of the top. You need to poke around and you will "feel" them disengage the tabs. Once this is done, the cluster slides right out.

 

To add, I dont remove the steering wheel. I take out the 2 bolts that hold it to the dash frame and let it rest in the seat out of the way.

 

Heck with that, just remove the whole dash and clean up all of the ducting to make sure there are no leaks. It would be a great time to install a new speaker in the passengers side.

 

...while you have the dash out, you might as well remove the seats and replace the carpet.

...while the carpet is out, you might as well remove everything else and do a full frame off restoration.

...ain't it funny how one little project gets going and the next thing you know the whole car is taken apart?

 

LIKE THIS

 

^^ Well almost like that (its a S30).

 

and THAT ^ started with a shift knob replacement!!

 

My needle bouncing and clicking has always been dry speed cables.

 

That fender well doesn't cut it LOL

 

And all these posts just show none of us know when to leave well enough alone.

Did I tell you about the time I needed to add air to the tires and next thing you know, I had the whole car stripped to the raw materials to make steel? Heck, trying to keep all that sand together that used to be glass was a real mess. Never could get that to fire back together again.

 

Just check if the cable came off lose from the back of the cluster.....??

 

^^^^hey, stop trying to put this thread back on track. You're killing the fun.

 

Mine 1979 280ZX is doing the exact same thing as described here.. Replaced long speedo cable, did not if.. I now understand it the gears and cluster need to come out, however, what is the fix when upper speedo cable is removed from cluster. Can gears be aligned?