We investigate non-standard Hamiltonian effects on neutrino oscillations, which are effective additional contributions to the vacuum or matter Hamiltonian. Since these effects can enter in either the flavor or mass basis, we develop an understanding of the difference between these bases representing the underlying theoretical model. In particular, the simplest of these effects are classified as "pure" flavor or mass effects, where the appearance of such a "pure" effect can be quite plausible as a leading non-standard contribution from theoretical models. Compared to earlier studies investigating particular effects, we aim for a top-down classification of a possible "new physics" signature at future long-baseline neutrino oscillation precision experiments. We develop a general framework for such effects with two neutrino flavors and discuss the extension to three neutrino flavors, and we demonstrate the challenges for a neutrino factory to distinguish the theoretical origin of these effects with a numerical example as well. We find how the precision measurements of neutrino oscillation parameters can be altered by non-standard effects alone (not including non-standard interactions in the creation and detection processes) and that the non-standard effects on Hamiltonian level can be distinguished from other non-standard effects (such as neutrino decoherence and decay) if we consider the specific imprint of the effects on the energy spectra of several different oscillation channels at a neutrino factory.