http://en.wikipedia.org/wiki/Precession
Gyroscopes work on "precession", see above.
They require both a rotating mass and a force to cause rotation about the axis.
While it all sounds sooooo complex, it is really pretty simple.
Get hold of a bicycle wheel (fronts are best, without the bike attached) and holding one end of the axel, wheel upright, spin the wheel. Now you will feel the wheel trying to turn around you.
Predicting the motion is also pretty easy. Taking your index finger (not that one... the first finger) imagine it as an indicator of right handed spin. That is the direction of a spinning object (wheel in this case) is clockwise from the finger's perspective. To generate a gyroscopic force, you need to apply a torque perpendicular to the axis of rotation. Use your left hand (index again) finger to indicate a torque in the same direction.
Now back to the bike wheel, holding the axel end in your left hand, spin the wheel with the right hand so it is rotating clockwise as you look at it. The force of gravity will apply a torque to the wheel to make it fall away from you...BUT... the gyroscopic effect will make the wheel turn into you and whack you on the nose.
You can use this to predict why a motorcycle (speedway bikes for example) will slide out with the back wheel when you push the bike hard down to the left or right. It also helps in understanding the effects of different engine rotation directions both in cars and bikes.
Appologies for the long discussion, but this shows the use of a gyroscope to dampen vertical vibration would be diffcult and complex. Definitly moveable balast.
The idea of a fluid tank is more likely a better way to go.
Ian P