In recent work in set theory the concept of "canonical structure" has emerged, in connection with combinatorial work on pcf theory. The idea is that there are many constructions that depend on the axiom of choice but, once realized, are actually independent of the specific choices made. Usually, this involves two steps: You construct an object, which is not quite canonical (say, a collection of subsets of a cardinal $\kappa$), but then you recognize that there is a natural ideal (say, the non-stationary ideal on $\kappa$) and the corresponding equivalence classes are canonical. Of course, by switching to a new model of set theory, the "canonical structure" may change, so sometimes one thinks of it as a sort of invariant of the models.
The first papers that explicitly mentioned the name "canonical structure" are by Cummings, Foreman, and Magidor, "Canonical structures in the universe of set theory", Parts I and II, Annals of Pure and Applied Logic 129 (2004), 211-243, and 142 (2006), 55-75.
The following quote is from the beginning of the introduction to Part I:
It is a distinguishing feature of modern set theory that many of the most interesting questions are not decided by ZFC, the theory in which we profess to work; to put it another way, ZFC admits a large variety of models. A natural response to this is to identify invariants which may take different values in different models, and which codify a large amount of information about a model.
Of particular interest are invariants which are canonical, in the sense that the Axiom of Choice is needed to show that they exist, but once shown to exist they are independent of the choices made. For example the uncountable regular cardinals are canonical in this sense.
Shelah discovered a large class of canonical invariants, the study of which he labeled PCF theory. These invariants include two which are central in this paper; Shelah [24, 26] (under some mild cardinal arithmetic assumptions on the singular cardinal $\mu$) defined two stationary subsets of $\mu^+$, the sets of good and approachable points. The definitions of these sets appear to depend on certain arbitrary choices, but (modulo the club filter) are in fact independent of these choices. Other canonical structures we study in this paper include the stationary sets of tight and internally approachable structures, and the collection of good points on a scale.
The two references cited in the quote are S. Shelah, "On successors of singular cardinals", in M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium ’78, pages 357–380, Amsterdam, 1979. North-Holland; and S. Shelah, "Cardinal Arithmetic". Oxford University Press, Oxford, 1994.
Besides the ongoing work by Cummings-Foreman-Magidor and Shelah, these ideas have been extended by others; Krueger and Ishiu come to mind.