Here is the snippet of code that will give the localRotation of a child relative to its parent if you have the 2 world rotations.

childObj.transform.localRotation = Quaternion.Inverse(childObj.transform.parent.rotation) * theChild.transform.rotation

You could convert the absolute rotation to a relative rotation by multiplying it with the inverse of the current rotation.

I'll be the first to admit my understanding of all of this is tenuous at best.

The following is my solution. Visually it seems to work as I want. The part I find slightly off is the conversion of the quaternion axis by the current base orientation. I did this on a whim to see what would happen. Oh and for some reason it won't rotate about the x-axis unless I update the z value of my quaternion?

public class Keyboard : MonoBehaviour {
    public GameObject 	objArm;
    public GameObject 	pivot;
    private Vector3 	  	objPosition;
    private Quaternion 	objRot;
    private Quaternion  lastRot = Quaternion.identity;

    private float rot = 0.0f;
    private int rotDir = 1;
    public float rotationIncr = 0.1f;

    private bool updateFromParent = false;
    ///Called from MouseLook.cs to tell player controller it's base orientation is in need of update
    void UpdateRotationFromParent(){
    	objArm.transform.rotation = pivot.transform.rotation;
    	objRot = objArm.transform.rotation;
    	updateFromParent = true;
    	RotateArm(lastRot);
    }
    // Use this for initialization
    void Start () {
    	Screen.showCursor = false;
    	Screen.lockCursor = true;
    	UpdateRotationFromParent();
    }

    // Update is called once per frame
    void Update () {
    	if (Input.GetKey ("escape")) {
    		Application.Quit();
    	}
    	UpdateArm();
    }
    void RotateArm(Quaternion q){
    	float angle = 0.0f;
    	Vector3 axis = Vector3.zero;

    	q.ToAngleAxis(out angle, out axis);

    	///Reorient axis of rotation to current object rotation
    	axis = objRot * new Vector3(axis.x,axis.y,axis.z);

    	//print("Rot: " + rot + " Axis: " + axis + " Angle: " + angle + " objRot: " + objRot);
    	objArm.transform.rotation = objRot;
    	objArm.transform.RotateAround(pivot.transform.position,axis,angle);
    }
    void UpdateArm () {
    	{	///Update rotation to simulate movement about Z axis from 0-90
    		rot += rotationIncr * (float)rotDir;
    		if(rot > 90.0f || rot < 0.0f){
    			rotDir *= -1;
    		}
    	}

    	///
    	Quaternion q = Quaternion.Euler(rot,0.0f,0.0f);
    	lastRot = q;
    	RotateArm(q);
    }
}