very simple Z Normal axis rotation new to Blender, so please forgive if obvious.I have been looking for an answer on this....

Goal: Simple object (e.g. cube) , once rotated 45 degrees, can have rotation on its Z Normal animated.  

I have tried using constraints, but not getting anywhere. I have tried parenting to an Empty. No luck.The object still wants to rotate on the Z of Global. Thank you for your help.
Let's see if I understand: you have a cube that is not aligned to the XYZ axes:



If you want to rotate it along its Local Z axis, then you only have to do R, then Z twice ("Rot: along local Z" appears in the status bar):



However if you want to animate the rotation of the object through keyframes, you first need to set its Rotation mode to "Quaternion":



That's the result with the default Rotation Mode ("Euler") and "Quaternion":