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I was reading Uniformization of Riemann Surfaces by Henri Paul de Saint Gervais (not a real person, but a group of French mathematicians), and the translator kindly points out that the name of "the greatest theorem of the 19th century" comes from the French word uniforme, meaning single-valued (as opposed to multi-valued). It may still seem obscure, but in the text (the introduction) it was explained pretty well.

For a lesser example, in the same introduction it talks about a group $\Gamma$ acting on the upper half plane (fixed point) freely and properly. That casual parenthetical remark cleared up the meaning of "free action" that I could never make sense of and had to look up its definition repeatedly.

I hope it is okay to ask for more examples of this sort. (You are welcome to rephrase the question.) It may not strictly be the original intention, but may have contributed to its wide acceptance but has since been forgotten (e.g. What is the naming reason of poles in complex analysis?). As other Terminology questions and answers show (especially in algebra), this may be quite tortuous to explain properly, and may not be more helpful than "just learn the definition."

For starters: what is so proper about a proper map?

Updated: To avoid being too vague that any terminology can have a story behind it (meromorphic, homology, etc), here are some guiding criteria:

1) Due to translation/importing (most often from French and German), or the multitude of meanings of the English word itself, the original meaning of the terminology has been lost in most texts on the subject;

2) it has been generalized out of the original context;

3) and that the original meaning helps in the understanding of the terminology, without having to give a long explanation.

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    $\begingroup$ I recall reading that Grothendieck introduced the terminology "nuclear space" into functional analysis to call attention to the development of nuclear arsenals during the Cold War. I'm not sure if this is actually true (and I can't substantiate it at the moment) but it suggests that some terminology may not be rooted in its underlying meaning. $\endgroup$
    – Paul Siegel
    Commented Aug 5, 2017 at 13:17
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    $\begingroup$ It may well be true, but don't "nuclear spaces" satisfy a théorème des noyaux ? This seems enough motivation for the name... $\endgroup$
    – Jean Duchon
    Commented Aug 5, 2017 at 15:32
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    $\begingroup$ Why don't you make the assumption that "nuclear" has been chosen because of its double sense? $\endgroup$
    – YCor
    Commented Aug 5, 2017 at 17:37
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    $\begingroup$ I'm not sold in this being the origin of the terminology free action. The action is free in the categorical sense : if T is a transversal to the orbits then any map from T to a G-set extends uniquely. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 5, 2017 at 18:27
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    $\begingroup$ This is rather embarrassing, but it was only recently that I realized the term "quiver" (in the sense of directed graph) was coined because it refers to a collection of "arrows" (directed edges) $\endgroup$
    – Sam Hopkins
    Commented Aug 16, 2017 at 13:31

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I thought "germ" would be a great example.

English not being my native language, I thought "germs" had to do with bacteria or microbes that cause diseases, which makes no sense in mathematics. I only realized it much later (though I'd like confirmation from others or the literature) that it had to do with the other, perhaps original, meaning: a portion of an organism capable of developing into a new one or part of one. One shall think of "to germinate", given the context of "stalks" and "sheaf/bundle". (There's also the English phrase "the germ of an idea", for what it's worth.)

It apparently was more widely used in function theory (of a complex variable), in which the infinitesimal neighborhood of any point in the domain of a holomorphic function determines the function completely (principle of analytic continuation). One may also find as formal definition of germ at a point as the sequence of Taylor series coefficients at that point. In general, of course, a sheaf in the modern sense may not be flabby/flasque, so that the germ of a section does not determine the section. (Speaking of sections, that's another terminology that may be confusing to some people.)

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    $\begingroup$ Grothendieck's choice of crystal is similaly motivated. (There is a quotation in Berthelot's book on crystalline cohomology explaining the choice) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 13, 2017 at 18:28
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    $\begingroup$ Yes, you should think of wheat germ. Which is part of a wheat stalk, many of which may be bundled together into a sheaf of wheat. (Let's not forget fibers.) Very agricultural, this terminology. In English, we also refer to a sheaf of paper, which is more suggestive of "horizontal" cross-sections (each given by an individual sheet of paper) than a bundle of "vertical" stalks. $\endgroup$
    – Todd Trimble
    Commented Aug 20, 2017 at 17:53
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Poincaré is usually said to be the first one to use homologous. He uses the term in his Analysis situs to mean the relation between manifolds that we nowadays refer to as cobordism or bordism. I have always interpreted his usage to be an extension of the relation of homology (or perspectivity) between triangles in geometry: one can think of a bordism from a manifold M to another N as a (very flexible!) perspectivity from one to the other.

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