It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb spectare (paradigm: specto, -as, -avi, -atum, -are), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (measure) experimentally.
Edit: after having a look to an online etymological dictionary, it seems the relevant Latin verb is another: spècere (or interchangeably spicere)= "to see", from which comes the root spec- of the latin word spectrum= "something that appears, that manifests itself, vision". Furthermore, spec- = "to see", -trum = "instrument" (like in spec-trum). Also the term "spectrum" in astronomy and optics has the same origin.
In algebraic geometry, I believe the term "spectrum", and the corresponding concept, has been introduced after the development of quantum mechanics became well known. In this context, the concept of spectrum as a space made of ideals is perfectly analogous of that in operator theory (think of Gelfand-Naimark theory, and that the Gelfand spectrum of the abelian C-star algebra generated by one operator is nothing but the spectrum of that operator).
I wouldn't be surprised if the term "spectral sequence" had something to do with "inspecting" [b.t.w. also "to inspect" comes from in + spècere...] step by step the deep properties of some cohomological constructions.
Maybe the term "spectrum" in homotopy theory and generalized (co)homology -but I don't know almost anything about these- has to do with "probing", "testing", a space via maps from (or to?) certain standard spaces such as the Eilenberg-MacLane spaces or the spheres. Does it sound reasonable?