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"
A lot of sources mention that the adjective "tropical" is given in honor of Imre Simon, but it seems hard to find who precisely coined the term. I found some sources which attribute this to some French mathematicians. Here is what Bryan Hayes writes on the topic:
For starters, what is that word “tropical” supposed to mean? Speyer and Sturmfels explain: “The adjective tropical was coined by French mathematicians, including Jean-Eric Pin, in honor of their Brazilian colleague Imre Simon.” Pin, in a 1998 paper (.pdf), deflects the credit to another French mathematician, Dominique Perrin, again noting that the name honors “the pioneering work of our brazilian colleague and friend Imre Simon.” Simon himself, in a 1988 paper (.ps), attributes the term to yet a third French mathematician, Christian Choffrut. Apparently, no one wants to lay claim to the word, and I can’t entirely blame them. Speyer and Sturmfels go on: “There is no deeper meaning in the adjective ‘tropical’. It simply stands for the French view of Brazil.”
Best Answer
You can find the original Lascoux & Schützenberger paper here. My French (especially mathematical French) is not great, so I haven't been able to determine how the term "plaxique" comes in. However, I can observe that L&S first introduce la congruence plaxique and define le monoïde plaxique as the quotient of the free monoid over the congruence. So, it seems to me that they were really thinking of the congruence as plactic/plaxic more than the monoid itself (perhaps a fine distinction?). They highlight the relevant properties of the congruence in Proposition 2.5, so maybe that provides a clue?
EDITED TO ADD: A quick scan of the OED yields no results for either "plactic" or "plaxic", but there is one result for the Latin "plaxus" under the etymology for the obsolete word "plash" (To bend down and interweave (stems partly cut through, branches, and twigs) so as to form a hedge or fence.):
So, perhaps "plaxique" is meant to invoke a sense of intertwining or weaving? I could see how that could apply to the congruence relation.
Bonus fun fact: Plaxico Burress makes an appearance in the OED in a citation for the entry "return date":