Are there plausible extensions of the Standard Model that could lead to early discoveries at the LHC? To address this general question on a concrete example, I will consider a class of minimal models with an extra massive neutral gauge boson Z'. I will first review different theoretical motivations for extending the SM gauge group with an extra U(1) factor, possibly broken near the TeV scale. I will then discuss the interplay between the bounds from electroweak precision tests and direct searches at the Tevatron, to identify the early LHC discovery potential. I will finally comment on the peculiar features of models where the Z' couples non-universally to lepton flavors and of string models with intersecting or magnetized branes.