Many of the toy gyroscopes come with a tower or pedestal. You may find that when spun up and the gyroscope is placed on the tower/pedestal, that the whole gyroscope rotates. E.g. the entire frame rotates.
The same may happen on some surfaces.
This is because of friction. The disk of the gyroscope is spinning fast. This applies forces/friction on the bearings. The bearings are naturally designed to be as friction free as possible. However the toy gyroscope bearings are crude and simple either plain bearings or very simple basic ball bearing. There is always some friction.
This results in torque on the cage. The cage will spin provided the friction between it and the tower or the surface it is placed on is less than the torque coming from the disk.
So how do you stop this from happening?
If you are putting it on a flat surface like a table. Put a matt under it. That could be a foam mouse matt. You could even use a towel. Maybe even a table cloth. This will reduce or stop it happening completely.
If you are putting on the tower/pedestal you need to reduce increase the friction between the tower and the gyroscope. You could add a small amount of putty.
Another thing you can do to reduce or stop this from happen is to oil the gyroscope. a tiny amount of light oil between the gyroscope shaft and cage will reduce friction on the bearings and hence less torque is created.