C#/Unity Camera rotation

It looks like you want the rotation to happen only when the right mouse button is held down. If this is true, then remove Input.GetMouseButton(1) from where it is now and then wrap it around transform.rotation = rotation;. It is as simple as that. FYI, you don't even need if (target). That is unnecessary but I leave it as it is.

public Transform target;
public float distance = 5.0f;
public float xSpeed = 120.0f;
public float ySpeed = 120.0f;

public float yMinLimit = -20f;
public float yMaxLimit = 80f;

public float distanceMin = .5f;
public float distanceMax = 15f;

private Rigidbody rigidbody;

float x = 0.0f;
float y = 0.0f;

// Use this for initialization
void Start()
{
    Vector3 angles = transform.eulerAngles;
    x = angles.y;
    y = angles.x;

    rigidbody = GetComponent<Rigidbody>();

    // Make the rigid body not change rotation
    if (rigidbody != null)
    {
        rigidbody.freezeRotation = true;
    }
}

void LateUpdate()
{
    if (target)
    {
        x += Input.GetAxis("Mouse X") * xSpeed * distance * 0.02f;
        y -= Input.GetAxis("Mouse Y") * ySpeed * 0.02f;

        y = ClampAngle(y, yMinLimit, yMaxLimit);

        Quaternion rotation = Quaternion.Euler(y, x, 0);

        distance = Mathf.Clamp(distance - Input.GetAxis("Mouse ScrollWheel") * 5, distanceMin, distanceMax);

        RaycastHit hit;

        if (Physics.Linecast(target.position, transform.position, out hit))
        {
            distance -= hit.distance;
        }
        Vector3 negDistance = new Vector3(0.0f, 0.0f, -distance);
        Vector3 position = rotation * negDistance + target.position;

        //Move Input.GetMouseButton(1) here
        if (Input.GetMouseButton(1))
        {
            transform.rotation = rotation;
        }
        transform.position = position;
    }
}

public static float ClampAngle(float angle, float min, float max)
{
    if (angle < -360F)
        angle += 360F;
    if (angle > 360F)
        angle -= 360F;
    return Mathf.Clamp(angle, min, max);
}