In some sense it was a failure of a certain diagram to commute that led to the Symmetric Spectra, S-modules, and other modern theories of spectra. Since these concepts underlie a lot of modern stable homotopy theory, everyone knows some version of this story. The category of spectra (not symmetric or S-algebras) goes back to Lima's 1959 paper The Spanier-Whitehead Duality in New Homotopy Categories, and is a natural construction if you want to do stable homotopy theory. Inverting the stable homotopy equivalences we get the stable homotopy category.
The "failure" I promised above is the failure of this category of spectra to have a symmetric monoidal structure. Such a structure was desired as a way to do more algebra in this setting (without it you have no hope of ring objects or modules over them). The diagram I mentioned which failed to commute was the diagram arising from the smash product on spaces, which the move to spectra did not preserve. For about 40 years it was thought that you could not have a symmetric monoidal category on spectra. See for instance the Lewis's 1991 paper Is there a convenient category of spectra? which shows that you can't have all the properties you want on such a category and also have it be symmetric monoidal. Thankfully, you can get enough of the properties you want and also get it to be symmetric monoidal. This was shown at the same time by two different teams of mathematicians:
- Elmendorf, Kriz, Mandell, and May created the category of $S$-modules
- Hovey, Shipley, and Smith created the category of symmetric spectra
Both are symmetric monoidal categories of spectra, have (different) desirable homotopy-theoretic properties, and both give the stable homotopy category when you invert weak equivalences. It turns out both approaches are equivalent in an even stronger sense than this, as can be seen for example in Schwede's S-modules and symmetric spectra.
[Disclaimer] This answer tells a story but may be missing important details or have things slightly wrong. That's because as a current graduate student I wasn't doing math at the time of these developments. So I'm glad this answer is CW so someone more knowledgeable can come and edit this if I got it wrong.
I think there's also a way to fit operads into this story, since every time I think of operads I think of $A_\infty$ and $E_\infty$ ring objects, which are ones where a key structural diagram (associativity and commutativity, respectively) does not commute on the nose, but it does commute up to homotopy. However, the coherence diagram doesn't commute up to homotopy, but does up to homotopies of homotopies. And for it's coherence diagram you need homotopies of homotopies of homotopies, etc. It seems to me that this arises from a similar goal as the above, namely to do algebra in stable homotopy theory. Before the issue was a lack of a product, but now the issue is that the product doesn't follow the rules (but it does up to infinitely coherent homotopy).