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?

**Edit:** The following paragraph from the wikipedia article on "primon gas" seems to support my guess:

"The connections between number theory and quantum field theory can be somewhat further extended into connections between topological field theory and K-theory, where, corresponding to the example above, **the spectrum of a ring takes the role of the spectrum of energy eigenvalues**, the prime ideals take the role of the prime numbers, the group representations take the role of integers, group characters taking the place the Dirichlet characters, and so on"