Say in my Hamiltonian I had a term
$$Q[a_j,a^\dagger_j]_\pm$$
where $Q$ is a constant. Suppose I didn't realise that this quantity equals 1 and calculated a normal order. Of course you get 0 However, the normal order of 1 should be 1 or else any operator could be multiplied by 1s and get 0 at the end. What is a correct way of normal ordering any operators?
Well, it depends on context. E.g.:
Assume we are given a classical model that we want to quantize, i.e. to construct a corresponding quantum theory. Be aware that quantization is not unique. The classical theory doesn't know about operator ordering so this introduce ambiguity. To parametrize our ignorance we should allow the possibility of an arbitrary constant term. Often the constant can later be fixed by other consistency requirements, cf. e.g. my Phys.SE answers here and here.
If we are just doing operator manipulations in some consistent operator formulation of a quantum theory, then there are no ambiguities. The commutation relations dictate the outcome of any operator rearrangement.
