How to stabilize angular motion (alignment) of hovering object?

You can use this script (C#) to stabilize your Rigidbody so it returns to an alignment where the local Y axis points upwards:

using UnityEngine; using System.Collections;

 public float stability = 0.3f;
 public float speed = 2.0f;

 // Update is called once per frame
 void FixedUpdate () {
     Vector3 predictedUp = Quaternion.AngleAxis(
         rigidbody.angularVelocity.magnitude * Mathf.Rad2Deg * stability / speed,
         rigidbody.angularVelocity
     ) * transform.up;

     Vector3 torqueVector = Vector3.Cross(predictedUp, Vector3.up);
     rigidbody.AddTorque(torqueVector * speed * speed);
 }

With the script you can control both how fast the Rigidbody stabilizes and how stable or wobbly the stabilization is.

Note: If you only want to stabilize along one axis, you can project the torqueVector onto that axis. For example, to stabilize along the local z (forward) axis only, do this before applying the torque:

torqueVector = Vector3.Project(torqueVector, transform.forward);