This article will cover the replacement of a water pump on a 1996 miata. This is generally done when the timing belt is being replaced. You should start off by following this link to a Timing Belt replacement. Then come back to this article to finish. It is always recommended to replace the water pump when the timing belt is replaced as it saves alot of labor. If you do not need to replace the timing belt you can use the plastic tie strap method which will make things go much faster. This picture shows the water pump pulley located in the center of the picture. The four bolts will need to be removed and the pulley taken off. The thermostat housing bolts (arrows) will not need to be removed.
This is the new water pump. You can look at it first to see what will need to be done. Below are the bolt holes that will need bolts removed from and the tensioner spring assembly.
There are also two bolts that need to be removed on the right side. You can see where there are in the picture below. Pay attention to the tensioner pulley and spring before it is removed. Removal and water pump replacement are not very difficult. There is a new gasket that is placed on the right side. To help hold it there and to help seal it, place a thin amount of silicone on both sides of the gasket.