Just finished working on a 1999 Mazda protege with the infamous "click click" starter issue at random. Everything checked out. While trying to test start the 5th time, the starter worked and in the middle just stopped dead. I tried to start it again and it worked fine. I diagnosed it as the starter relay. NOT THE SOLENOID, THE RELAY!! The web is full of so called expert mechanics telling people the relay is located on top of the starter (that's the solenoid).
But where is it? It is a mystery all over the web with many wrong answers if any. It is like it is some kind of secret. In this case the relay was located in the plastic fuse box on the driver's side in the front of the engine compartment (Says FUSE BOX on top). Most Mazdas from 1990's to 2005 (+?) have the starter relay located in the black plastic engine compartment fuse box.
The relay you are looking for is part number B5B4 and it is written on the side of the relay. This relay is usually easy to spot and is mostly situated with three other large relays since it is a high amp relay. The starter relay may be the tallest of them. In this particular car the OEM replacement part ( $14.95 Rock Auto Intermotor made in Japan) was the same color as the old part and was easy to spot (bright yellow green). I don't know if this is the second replacement I did or if they came that color from the factory. It's not like my place is full of Mazdas so I could check others. Install the relay with a little dielectric grease on the copper connections on the relay as the factory does. Installed new relay, starter seems to work better and spin faster, no issues. Out the door went the car hopefully to not come back with the same problem. Intermittent electrical problems in a car are the hardest problems to diagnose.
Good luck.