I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less.
I might want to understand bordism, and start by trying to understand submanifolds, then realize that this is really hard to do and try instead to handle a combinatorial approximation. Then I might define simplicial homology.
After dealing with simplicial homology for a few decades, I might tire of my confinement to the simplicial setting, but might nonetheless want to reason combinatorially about simplices, and I might then define the singular simplices functor and worry about singular homology.
Motivated by Stokes's theorem and Poincaré duality, I might have the idea that Grassmann's differential forms could be considered as dual to smooth submanifolds in some sense, and I might define de Rham cohomology on manifolds.
Once I knew about the Mayer–Vietoris sequence and had started to get a feeling for of local–global relations in (co)homology theories, and in particular knew Poincaré's lemma, I might decide it was a good idea to try and understand (co)homology in terms of the combinatorics of a cover of contractible open sets, and I might eventually just define cohomology as the direct limit of a set of algebraic structures derived from covers. This would also have benefit of smoothing out irregularities in my object space.
Thinking about the properties of the de Rham complex in terms of supports of differential forms and still keeping the Poincaré lemma in mind, I might also define fine sheaves and ultimately cohomology with coefficients in a sheaf, if, for example, I were exceptionally creative and trying very hard not to look like an analyst while imprisoned by the Nazis in a POW camp.
On the other hand, I've looked at Dieudonné's history and the original papers of Alexander and Spanier, but I still have no real idea what would inspire me to define Alexander–Spanier cohomology. Does anyone have any insight?
P.S. [7 Dec.]: Massey has an account in his essay "A history of cohomology theory" in the collection History of Topology (ed. Ioan James). On p. 567, he states
It is not difficult to see why Whitney and the other participants at the Moscow conference must have been mystified when Kolmogoroff and Alexander wrote down their definitions of a product of cochains. These definitions were pure ad hoc formulas, presented with no motivation. It is hard to guess how Alexander and Kolmogoroff arrived at them. It must have seemed like numerology or magic.
I've learned from Massey's account that Alexander(–Kolmogorov!)–Spanier cohomology was likely intended to be dual to Vietoris homology but not exactly how this duality functioned. Vietoris homology was initially defined, as I understand, on compact metric spaces, with simplices ordered sets of points within an $\epsilon$-neighborhood, and $\epsilon$ taken to zero, with cycles being sequences of cycles modulo eventual boundaries. While this approach to zero is reminiscent of modding out functions vanishing on a neighborhood of the diagonal, I still do not know their motivation for doing so.