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.