the starter
Possibly, depending on what causes the blackouts.
starter motor need overhaul
Loose connection? Bad starter (flat spot)? Bad starter solenoid?
First I would check that all the wires are hooked up correctly. Next check the battery cables and the ignition fuse. Next I would check the solenoid. Good luck
dead battery
Is the SLEEP timer on? Go into the menu and check.
Starter misaligned? Starter engaged with engine running? Metal fatigue?
you can find information on a starter of early miatas on: http://www.racecaddy.com/mx5/tech/starter-mx5.html They have location, causes of failure and how to change it
If it is installed incorrectly and causes the starter to be engaged continuously/always running it would kill the starter fairly quickly.
It can be a defective starter assembly that causes this problem. There are two parts to a starter assembly: the starter motor and the starter solenoid. The offending culprit is usually the starter solenoid, and it should be replaced.
decribe two causes of higher than normal starter current draw