Models of new physics that introduce new particle species or new couplings (types of interactions beyond the familiar electromagnetic, weak, and strong) can lead to an observable rate of muon-to-electron conversion, and there are loads of such models in the wild. New particles could come about due to supersymmetry, the presence of a heavy neutrino, or an extended Higgs sector (i.e., more Higgs particles than are absolutely necessary for electroweak symmetry breaking). New heavy bosons could lead to new interactions that we haven't yet probed. (This is analogous to how the weak force is negligible at low energies: it is mediated by heavy gauge bosons. If there are even heavier ones out there, perhaps they lead to additional forces that we have to work hard to detect.) "Compositeness" is the idea that elementary particles aren't actually elementary, and the heretofore unnoticed substructure could lead to anomalous interactions across lepton families. "Leptoquarks" is a generic term for anything that allows the conversion of leptons to quarks, and vice versa, so leptoquarks could evade the approximate lepton flavor conservation in the Standard Model.
All these and more could lead to muon-to-electron conversion. If a signal is observed, it will be Earth shattering, but it will also be difficulty to pick apart what sort of new physics is the cause using just the Mu2e experiment.
