I can post studies all day that show that our largest natural GH pulse is overnight...Its a fact. Its also a fact that administering rHGH will inhibit any natural GH from being secreted through negative feedback (rHGH will raise IGF levels which will stimulate somatostatin secretion which will inhibit Growth Hormone Releasing Hormone, amongst other things, which will inhibit natural GH secretion etc etc).
The reason why your package insert directs one to administer at night is because rHGH is prescribed only for GH deficiency. If you're not producing any GH naturally, you should take the rHGH pre-bed to "mimic" the natural GH pulse that occurrs with GH-healthy people.
As we enter our "golden years", GH secretion reaches a point in which it is immeasurable...at that point, bed time dosing would be beneficial.