Suppose your robot walks vertically on two legs, and you want to mount a gyroscope in the center of the robot with a vertical spin axis.
As a seat-of-the-pants engineer, I would ask how much rolling moment is needed to keep the robot from falling very far before it places a foot so it will stop falling.
This depends on the robot's mass, how high it is, how far apart the legs are, and how quickly they move.
When you've decided on the maximum moment, consider how much precession you can tolerate.
Suppose the gyro consists of mass $m$ concentrated in a ring of radius $r$ spinning clockwise at angular velocity $w$ (radians/sec) when looked at from above.
Suppose you can pitch the vertical axis of the gyroscope forward or backward.
If you move the vertical axis forward, that will have the effect of creating a gyroscopic reaction trying to roll the axis to the right (and vice-versa).
If $v$ is the angular velocity at which you pitch the axis forward, the roll moment will be:
$m r^2 w v$
The amount by which you pitch the axis forward depends on how long you need to apply the moment.
BTW, you can also apply a fore-and-aft moment by rolling the axis left or right.
Note: Anti-rolling gyro.
ADDED in response to comment:
OK, so you're concerned about pitch angle. The gyro axis could be in any direction except left-to-right. In order for the gyro to work, it needs to be free to precess. In some of the ship stabilizer applications, it simply has dampers (springs & shock absorbers) holding the axis in place, but giving it freedom to move.
Here's how I think about it.
Suppose in front of your segway there was a stream of water from a fire hose being sprayed from left to right.
You could, holding a flat plate in your hand, put it in the water stream in such a way as to deflect the water down or up at an angle.
That would provide a corrective force that you could use to balance your segway.
You can work out the physics of how much force you could get as a function of the deflection angle and the amount and speed of the water.
Now instead of a stream of water, you have a stream of metal.
Instead of going in a linear stream, it's going in a circle.
Instead of simply being deflected downward at an angle,
it's axis of rotation is changing so as to deflect the direction of the metal downward on the side where it's moving left-to-right (and upward on the return trip).
So you need to get an estimate of how much corrective force you want to be able to get, and how much axis-movement you want to pay for to get it.
I would experiment, first trying something like a bicycle wheel, maybe filled with water, maybe put heavy metal weights around the outside.
Spin it up by some means (cord, electric drill, or direct motor) and measure the effect.
You can double the effect by doubling the mass or doubling the rotation rate.
You can quadruple the effect by doubling the radius.