Big Red Button
Some of this material is based on Sean Archer's writeup. I made a couple of small modifications, but it's basically the same setup.
What we need to do is cut the power to the accessories and complete the circuit that runs the starter. Sounds easy, right? Well, actually - it is. Both of these circuits run through the ignition switch. IGN2 is the accessory circuit and ST is the starter. Again, see Sean's site for some diagrams.
We'll need a relay to handle the power reliably. You can either use the dual pole relay (Siemens VF4-45F11) or a couple of inexpensive, easy to find single pole auto relays available at Radio Shack or Pep Boys. Using the single pole relays does complicate the wiring somewhat, but not too badly. One relay is used for each circuit.
Here's a rough diagram of how to connect the two relays. There's a big white connector that plugs into the ignition switch that you need to unplug. It's got 6 wires on it. The two that interest us are black with blue stripe (ST) and black with red stripe (IG2). The IG2 circuit provides power for the relays. If you use a double-pole relay, your wiring will look like Sean's version. All the new wiring is in green, and the stock wires are in the appropriate colours. I've heard one or two reports of the ignition switch connector changing a bit or of people getting confused by the diagram - use the wire colours to identify the wires, not the positions in the connector.
NOTE for 1999 and later Miatas: The wiring colours have changed. ST is white with a black stripe. IG2 is still black with red stripe, but don't get confused by the red wire with the black stripe that's nearby.
It's possible to do this without cutting a single wire. Here's how.