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A small one. Arduino Micro, Pro Mini or Teensy 3.2
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You cannot get "high accuracy" in consumer-grade equipment. The real military stuff is expensive and you must sign stuff to say you're not going to export it.
However, the common consumer-grade stuff, like the sensors in a mobile phone, are quite adequate for this job. The MPU6050 or MPU9250 and their brothers are really very good for this.
- A common RC hobby servo will be quite adequate. This is exactly the type of job that they are designed for. Just make sure that you put it on the aerodynamic balance point of the fin, to minimise the torque required.
The main problem is going to be vibration-hardening all the components. You can't rely on a breadboard. The initial acceleration of the rocket off the pad will pull chips out of their sockets. Get it all soldered together on proto-board and never let a wire just hang off a solder joint - always clamp it down to the board with a cable tie or something.
Don't throw away simple analog control. You can get extremely good results with 4 light-dependent-resistors (LDRs) aimed at the horizon. When one sees more bright sky than its counterpart on the other side, it can move the fins to make the rocket perfectly vertical.