I am about to indulge in a bit of philosophical rambling. Some of this will eventually make it onto the article page, when I get it coherent in my mind. Please comment as you like. 

 "Making the suspension invertible" is indeed a very common motivation in texts and courses for the introduction of the objects we call spectra. But to me, this is less convincing than representability of all cohomology functors (see Brown representability theorem). The entire topic is difficult to treat. As I imply in my earlier comment, there are many different constructions of various categories, all of which purport to be "the category of spectra" or "the stable homotopy category." Adams' construction is described in the article, and I think that this is still the model that most homotopy theorists have in mind. But Boardman's model has better formal properties. The really cutting edge categories are better still (there are two that I'm aware of: E-infinity ring spectra and S-modules) because they are model categories with good properties before passage to homotopy, most notably a smash product that commutes on the nose. But do we really want to talk about any of this in the article? My worry is really that it is misleading to define a spectrum as is done in the article, from the modern viewpoint; but I can't see that any alternative is really any better. 

 On another note, I think the comparison to derived categories might be delivered with less apology. There is quite a bit of current research happening along those lines. For example, there are various sets of axioms one might use to define a stable homotopy category, e.g., triangulated, closed symmetric monoidal, a set of categorically small generators, etc. With suitable choices the derived category satisfies these. See the monograph of Hovey, Palmieri, and Strickland, Axiomatic Stable Homotopy Theory. Moreover the model categories approach applies here as well. Consider the category of bounded chain complexes of modules. The cofibrant objects in the usual model structure are the perfect complexes, the ones with each term projective (I think?). So projective resolution is cofibrant replacement, and this is actually well-defined up to unique isomorphism (I think?) in the derived category. 

 Any thoughts?