This question was asked four years ago, but never got an answer (though, it had some discussion in the comments). This question could have also been asked on academia.SE but it's not clear it's really on topic there either. Let me try to answer so that this doesn't linger forever on the unanswered queue.

With any question about writing, in addition to the guiding principle of not inventing new terminology for something that already has standard terminology, another good guiding principle is do what's best for your reader. Having prepared thousands of pages of writings in my research and teaching, I've found that what's best for the reader is usually whatever is clearest. Hence, when I make a construction in one of my papers and want to refer to it elsewhere in the paper, I give it a numbered environment, just like an equation, definition, lemma, proposition, theorem, conjecture, etc.

Construction 2.1: Under the assumption that there are only finitely many primes $p_1, \dots, p_k$, let $P = p_1p_2\cdots p_k + 1$.

Later in the paper, maybe I need to do the same trick again, and can say "Using the same technique as Construction 2.1, we now ..."

I've also had things like:

Standing Hypothesis 1.1: We assume all model categories are cofibrantly generated in this paper.

or

Agreement 2.1: Let us agree that by "operad" we mean "reduced operad" in what follows.

I think this is MUCH clearer than words like Scholium and Ansatz, that have the potential to throw off readers who have not seen words like that before. I teach so many students from Asia for whom English is already a second language, and words that draw from yet a third language tend to throw them off quite a lot.

Now suppose you're teaching a course on proof-writing, or writing a paper about how we write mathematics, and want a way to refer to the general idea of a construction done inside a proof. In that case, I think "gadget" is a good choice, even though it has a technical meaning elsewhere, because most students/readers know what a gadget is, especially if I mention the real-world meaning of the term and why I chose that word for this concept. Based on the comments, it sounds like there is no standard term. If you want to avoid "gadget" because of the connection to computational complexity, you can use "gizmo" instead.

Lastly, I want to point out that one way to avoid gizmos in proofs is to create lemmas, following Terry Tao's advice. You could imagine doing all your writing in such a way that, wherever you were constructing something inside a proof, you deliberately pulled that out into a lemma like:

Lemma 3.1: If $p_1, \dots, p_k$ is a finite list containing all prime numbers, then $P = p_1\cdots p_k + 1$ is also prime.

This technique of writing is extremely clear and helps the reader focus on one proof at a time, instead of a proof within a proof. Incidentally, this gives me another idea. You could use "inception" to refer to the phenomenon of proofs/constructions within existing proofs of other statements.

This question was asked four years ago, but never got an answer (though, it had some discussion in the comments). This question could have also been asked on academia.SE but it's not clear it's really on topic there either. Let me try to answer so that this doesn't linger forever on the unanswered queue.

With any question about writing, in addition to the guiding principle of not inventing new terminology for something that already has standard terminology, another good guiding principle is do what's best for your reader. Having prepared thousands of pages of writings in my research and teaching, I've found that what's best for the reader is usually whatever is clearest. Hence, when I make a construction in one of my papers and want to refer to it elsewhere in the paper, I give it a numbered environment, just like an equation, definition, lemma, proposition, theorem, conjecture, etc.

Construction 2.1: Under the assumption that there are only finitely many primes $p_1, \dots, p_k$, let $P = p_1p_2\cdots p_k + 1$.

Later in the paper, maybe I need to do the same trick again, and can say "Using the same technique as Construction 2.1, we now ..."

I've also had things like:

Standing Hypothesis 1.1: We assume all model categories are cofibrantly generated in this paper.

or

Agreement 2.1: Let us agree that by "operad" we mean "reduced operad" in what follows.

I think this is MUCH clearer than words like Scholium and Ansatz, that have the potential to throw off readers who have not seen words like that before. I teach so many students from Asia for whom English is already a second language, and words that draw from yet a third language tend to throw them off quite a lot.

Now suppose you're teaching a course on proof-writing, or writing a paper about how we write mathematics, and want a way to refer to the general idea of a construction done inside a proof. In that case, I think "gadget" is a good choice, even though it has a technical meaning elsewhere, because most students/readers know what a gadget is, especially if I mention the real-world meaning of the term and why I chose that word for this concept. Based on the comments, it sounds like there is no standard term. If you want to avoid "gadget" because of the connection to computational complexity, you can use "gizmo" instead.

Lastly, I want to point out that one way to avoid gizmos in proofs is to create lemmas, following Terry Tao's advice. You could imagine doing all your writing in such a way that, wherever you were constructing something inside a proof, you deliberately pulled that out into a lemma like:

Lemma 3.1: If $L = \{p_1, \dots, p_k\}$ is a finite list of prime numbers, then $P = p_1\cdots p_k + 1$ is divisible by a prime not in $L$.

This technique of writing is extremely clear and helps the reader focus on one proof at a time, instead of a proof within a proof. Incidentally, this gives me another idea. You could use "inception" to refer to the phenomenon of proofs/constructions within existing proofs of other statements.