Gyroscopes and Conservation of Angular Momentum Let's begin by insisting that angular momentum is conserved everywhere and at all times. Since it is a vector quantity, direction matters. 
Consider a gyroscope on the end of shaft, the other end of which sits on a free pivot point. After being spun up the gyroscope is dropped so that it begins to precess around a vertical axis centered at the pivot.
The precession is caused by a torque from gravity acting on the spinning mass of the gyroscope. The energy involved in precession has its origin in the kinetic energy in the spinning gyroscope. So far so good.
For the angular momentum of the precession (not of the gyroscope wheel itself) to be preserved, there must be an equal and opposite pointing angular momentum somewhere in the system.
Where is it?
 A: 
The energy involved in precession has its origin in the kinetic energy in the spinning gyroscope. So far so good.

No. 
You spin the gyroscope (upright) so that it has an angular velocity $\omega$. The kinetic energy associated with this motion is:
$$ \frac{1}{2}I_{\mathrm{axle}}\omega^2,$$
where $I_{\mathrm{axle}}$ is the moment of intertia of the gyro about its axle.
The gyroscope then falls a bit under gravity and then it starts precessing. The angular velocity of the gyroscope about its axle is still $\omega$ -- assuming a point contact with the ground and no air resistance, there was no torque in that direction that could have reduced it.
The gyroscope then starts precessing about the $z$ axis (gravity) with angular velocity $\Omega$. This introduces a kinetic energy $$ \frac{1}{2}I_z \Omega^2,$$
where $I_z$ is the momentum of intertia of the gyro assembly about the $z$ axis, not about its axle.
This kinetic energy, not originally present in the system, comes form the gravitational potential energy $mgL(1-\cos\theta)$ lost when the gyro drops by angle $\theta$ before precessing.

For the angular momentum of the precession (not of the gyroscope wheel
  itself) to be preserved, there must be an equal and opposite pointing
  angular momentum somewhere in the system.

Angular momentum (about an axis and/or a point) is only conserved in a closed system, i.e. a system which no external torques act upon.
For the $\omega$-associated angular momentum, that's still conserved as said before, there are no torques tangential to the gyro that can slow down its rotation.
For the total angular momentum, about (say) the point of contact, gravity provides the external torque. 
Hence you expect a change in angular momentum:
$$ \Delta \mathbf{L} = \Gamma \Delta t = (\mathbf{r}\times m\mathbf{g})\Delta t,$$
which is exactly what causes precession.
A: "Conserved" doesn't mean "unchanging".  It means "only changed by inputs and outflows".  
In the case you mentioned, gyroscopic precession due to gravity, gravity is exerting a torque on the assembly.  That's what causes the changing angular momentum as the assembly rotates.
