The discussion in the comments has triggered my curiosity: I've done a little bit of research and I almost immediately stumbled upon the [Wikipedia entry on the Peano-Jordan measure](https://en.wikipedia.org/wiki/Peano–Jordan_measure).  There I found exactly the terms "*simple sets*" and "*polyrectangles*" used as synonyms for the unions of finite families of $n$-rectangles, i.e. for the set
$$
S=\bigcup_{i=1}^q C_i\quad q\in\Bbb N_{>0}
$$
where $n$ is the dimension of the Euclidean $\Bbb R^n$ considered and $C_i$ are $n$-rectangles defined as $C_i=[a_{1i}, b_{1i})\times\cdots\times[a_{ni}, b_{ni})$. However, I was not able to find where these sets are called in this way within the references stated in the relevant section of the entry: thus I started searching in my personal library. I found the locution "*plurirectangles*" (*plurirettangoli* in Italian) in the classical monograph \[1], chapter II, §2.7, p. 119, as shown in the picture below:

[![enter image description here][1]][1]

This makes me sufficiently confident in proposing the adoption of this, somewhat traditional, terminology (i.e. plurirectangle, see the discussion in the [comments by Pietro Majer](https://mathoverflow.net/questions/472097/is-there-a-name-for-finite-unions-of-intervals/472170?noredirect=1#comment1226826_472170) and [Daniele Tampieri](https://mathoverflow.net/questions/472097/is-there-a-name-for-finite-unions-of-intervals/472170?noredirect=1#comment1226835_472170)).

**Addendum**. As I customarily do, when I try to be faster sacrificing precision I end up being at best badly inaccurate. The standard locution for finite unions of $k$-intervals *is pluriintervals* (or *plurintervals* if you prefer to translate so the Italian word *plurintervalli*): both Gaetano Fichera in his only book ([1a], chapter I, §I.12, p. 29 footnote (9)) and Carlo Miranda in his last (and recent) treatise ([2a], chapter II, §36, p. 160) use exactly this word, as shown in the following pictures:
[![enter image description here][2]][2]

[![enter image description here][3]][3]

Possibly, this settles the problem.

**Reference**

\[1] Federico Cafiero (1959), *Misura e integrazione* [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, [MR0215954](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0215954), [Zbl 0171.01503](https://www.zbmath.org/?q=an%3A0171.01503).

**Addendum references**

[1a] Gaetano Fichera, *Lezioni sulle trasformazioni lineari. I.: Introduzione all’analisi lineare (Lectures on linear transformations, I.: Introduction to linear analysis)*, (Italian) (third reprint, 1962) Trieste: Istituto Matematico dell'Università, pp. XIX+502 (1954), [MR67346](https://mathscinet.ams.org/mathscinet-getitem?mr=67346), [Zbl 0057.33601](https://zbmath.org/0057.33601).

[2a] Carlo Miranda, *Istituzioni di analisi funzionale lineare. Volume I (Foundations of linear functional analysis. Volume I)*, (Italian) Unione Matematica Italiana. Bologna: Pitagora Editrice, pp. iii+596 (1978), [Zbl 0697.46001](https://zbmath.org/0697.46001).

[2b] Carlo Miranda, *Istituzioni di analisi funzionale lineare. Volume II. (Foundations of linear functional analysis. Volume II)*, (Italian) Unione Matematica Italiana. Bologna: Pitagora Editrice. pp. 597-748 (1979), [Zbl 0697.46002](https://zbmath.org/0697.46002).


  [1]: https://i.sstatic.net/Wi7VtrVw.jpg
  [2]: https://i.sstatic.net/jNBCAlFd.jpg
  [3]: https://i.sstatic.net/oFY29YA4.jpg