Miatas are known for speedometer cable noise and associated needle wobble. The obvious symptom is a loud clicking sound that varies with speed, and a bouncing or wobble of the speedometer needle when driving. You may find it is worse when hot, or when cold.
The standard fix is to simply replace the cable. The cable is relatively inexpensive ($80), and can be changed fairly easily. If you check the regular Miata forums, you will hear that it is impossible to lubricate an existing cable. I will admit that buying a new cable is a sure fire way to fix, and that most lubrication efforts fail. However, if you want to give it a try, this is what I did that worked.
First, disconnect the speedometer cable connection from the transmission end. Carefully unhook it from the two tie downs that hold the cable against the transmission tunnel. Then gently pull the cable into the engine compartment. Inside the connector is a round pin with a tang on it that spins. This drives the cable that goes to the speedometer. The pin is in a small little cup. What I did is strap the cable vertically so that you can fill the cup with oil, and let it slowly drain into the cable sheathing. I used a small amount of used synthetic motor oil. I filled the cup up about fives times over 30 minutes. Take your time, I think the longer you leave the cable end higher than the speedometer, the better your results.
Give it a try and let me know how it goes!