To my way of thinking, the other answers are missing an important element, a necessary feature for a mathematical tool or method to be called "trick."
Namely, in order to be called a "trick," a method or technique must involve artifice of some kind. When we treat a mathematical object as something that it isn't really, or when we pretend that something is other than it is, in order to advance an argument, we are using trickery. (I am not suggesting that the mathematics is not correct.) When we solve a problem by placing our focus on something else, in which we aren't actually interested as such, and which may even be silly in some way, but by doing so become successful in the original problem, then we are using trickery. When we replace a robust concept, in which we are really interested, with a modified version of it, perhaps even an absurd version of it, but which makes the argument work, then we are using trickery.
For example, with Craig's trick, we replace a formula $\varphi$ with the conjunction with itself $\varphi\wedge\varphi\wedge\dots\wedge\varphi$ repeated many times over. Although this new assertion is logically equivalent to $\varphi$, nevertheless we don't actually care about this new assertion as such, and indeed it is absurd. We use it merely to code some information: the number of times it was repeated. By this artifice, we can code information into an axiomatization or presentation. Thus, every computably enumerable theory has a computable set of axioms. The same idea works in group presentations: every c.e. presentable group has a computable presentation.
With Scott's trick, the issue to be solved is that the equivalence class of an object forms a proper class, which can cause certain problems, and so we replace that equivalence class with the set of rank-minimal members of the class. If we think of this fake equivalence class as the real thing, then everything works great! This trick is surprisingly robust, and can be used to find small canonical sets of representing structures in almost any situation. For example, in ZFC there is a definable manner of choosing a set of groups from each group isomorphism class: the rank-minimal groups from that class. This is a trick, because we don't really care much about that particular collection as such.
With Rosser's trick, we replace the concept of a theory $T$ proving a sentence $\sigma$, with: $T$ proves $\sigma$ by a proof for which there is no shorter proof of $\neg\sigma$. When you think of "proof" using this concept, then Gödel's incompleteness theorem is improved to the Gödel-Rosser theorem, where one can drop Gödel's extra hypotheses about $\omega$-consistency. This is a trick, because we don't actually want to think about "proof" using Rosser's concept, except that it makes the argument work.
In each case, we replace the concepts or objects in which we are truly interested by concepts or objects that we don't actually care about as such and which in several cases are comical versions of the original, except that they make the argument work.