A quick GOOGLE search revealed the following material on the subject:
electriciantraining.tpub.com/14187/css/14187_139.htm
A gyro can have different degrees of freedom, depending on the number of gimbals in which it is supported and the way the gimbals are arranged. Do not confuse the term "degrees of freedom" with an angular value such as degrees of a circle. The term, as it applies to gyros, is an indication of the number of axes about which the rotor is free to precess. A gyro mounted in two gimbals has two degrees of freedom.
When two gimbals are used, the gyro is said to be UNIVERSALLY MOUNTED. This arrangement provides two axes about which the gyro can precess. These two axes and the spin axis intersect at the center of gravity of the entire system (excluding the support). Because of this arrangement, the force of gravity does not exert a torque to cause precession. The rotor, inner gimbal, and outer gimbal are balanced about the three principal axes.
Elmer A. Sperry - Degrees of Freedom
A gyro has two basic mountings: one with three degrees of freedom, and one with only two degrees of freedom. A gyro having three degrees of freedom is able to rotate about three axes including its own axis of rotation. Foucault used such a model to illustrate the rotation of the earth, crafting a model with a gyro mounted in two rings. One ring rotated on a horizontal axis within the other, which in turn rotated on a vertical axis.
http://www.navymars.org/national/tra.../14187_ch3.pdf
DEGREES OF FREEDOM
A gyro can have different degrees of freedom, depending on the number of gimbals in which it is supported and the way the gimbals are arranged. Do not confuse the term "degrees of freedom" with an angular value such as degrees of a circle. The term, as it applies to gyros, is an indication of the number of axes about which the rotor is free to precess.
A gyro mounted in two gimbals has two degrees of freedom. When two gimbals are used, the gyro is said to be UNIVERSALLY MOUNTED. This arrangement provides two axes about which the gyro can precess. These two axes and the spin axis intersect at the center of gravity of the entire system (excluding the support). Because of this arrangement, the force of gravity does not exert a torque to cause precession. The rotor, inner gimbal, and outer gimbal are balanced about the three principal axes.
TWO DEGREES-OF-FREEDOM GYROS
The two-degrees-of-freedom (free) gyros can be divided into two groups. In the first group, the gyro's spin axis is perpendicular to the surface of the Earth. Thus the gyro's rotor will spin in a horizontal plane. These gyros are used to establish vertical and horizontal planes to be used where stabilized reference planes are needed.
In the second group, the gyro's spin axis is either parallel to the surface of the Earth or at some angle other than perpendicular. The spin axis of the gyro in the gyrocompass, for example, is maintained in a plane parallel to the surface of the Earth It is aligned in a plane of the north-south meridian. Once set, it will continue to point north as long as no disturbing force causes it to precess out of the plane of the meridian.
en.wikipedia.org/wiki/Gyroscope
Within mechanical systems or devices, a conventional gyroscope is a mechanism comprising a rotor journaled to spin about one axis, the journals of the rotor being mounted in an inner gimbal or ring; the inner gimbal is journaled for oscillation in an outer gimbal for a total of two gimbals.
The outer gimbal or ring, which is the gyroscope frame, is mounted so as to pivot about an axis in its own plane determined by the support. This outer gimbal possesses one degree of rotational freedom and its axis possesses none. The next inner gimbal is mounted in the gyroscope frame (outer gimbal) so as to pivot about an axis in its own plane that is always perpendicular to the pivotal axis of the gyroscope frame (outer gimbal). This inner gimbal has two degrees of rotational freedom.
The axle of the spinning wheel defines the spin axis. The rotor is journaled to spin about an axis, which is always perpendicular to the axis of the inner gimbal. So the rotor possesses three degrees of rotational freedom and its axis possesses two. The wheel responds to a force applied about the input axis by a reaction force about the output axis.
en.wikipedia.org/wiki/Turn_and_balance_indicator
Turn indicator
The turn indicator is a gyroscope instrument that works on the principle of precession. The gyro is mounted in a gimbal. The gyro's rotational axis is in-line with the lateral (pitch) axis of the aircraft, while the gimbal has limited freedom around the longitudinal (roll) axis of the aircraft.
MY COMMENTS
The existence of the spin axis itself does not provide a degree of freedom because the spin axis cannot precess. But each gimbal enables the spin axis to precess in one additional direction. So the number of degrees of freedom is equal to the number of gimbals.