Gyroscopes Output Voltage drift when not moving, trivial question
The best way to answer that question is to take a look in the datasheet for your chosen gyro.  You didn't say which one you wanted to use, so let's take a look at a reasonably good one, the ITG-3200.

According to the datasheet:



The zero rate output can vary by as much as 40 degrees per second, and can vary by this much over its rated temperature range.

It seems like a lot, but it's only 2% of its full scale range of 2000 degrees per second. Furthermore, the device actually contains a temperature sensor. Using the sensor, you will be able to calibrate the device to account for the error due to temperature.

Put your device in the freezer over night. Take it out, and place it on a table, As the device warms up, take readings of the temperature and each of the axis, and build up a table which relates the temperatures to the axis readings. Store enough of those values in the Arduino code so that you can compensate for temperature in the future.



Also, 2000 degrees per second might well be much much higher than you want for your application. So my advice would be to choose the gyro which is rated for the lowest maximum rate that you wish to measure in your application. Then you will be able to know it's static to within about 2% of the full scale rate. But whichever device you choose, check the datasheet carefully first.