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The are some parts of math in which you encounter easily new structures,
obtained by modifying or generalizing existing ones. Recent examples
can be tropical geometry, or the theory around the field with one element. If one works in those areas, one cannot avoid the problem of naming new objects.

When working with such a "new" notion, more general than an existing
one, you have different options to name it. Either the red herring
option, like group without inverses, a brand new name, like monoid, a derived name, like semigroup (which is actually a group without inverses and
without an identity element), or no name at all, so a very long name,
i.e. the category $M$ of sets with an associative binary operation.
Or even you can also decide to use the old name with a new meaning.
(The examples I wrote don't pretend to have a historical justification).

Although the red herring construction is used everywhere in math, I
feel that it is not a good practice. To use the old name with a
different meaning can be the origin of a lot of errors. And the
option of not giving any name at all is like you elude your
responsibility, so if someone needs to use it they will have to put a
name to it (maybe your name?).

So my preferred options are to choose a derived name or a new name.
Derived names are quite common: e.g. quasicoherent, semiring,
pseudoprime, prescheme (which is an old term), and they
contain some information which is useful, but sometimes they are ugly,
and it could seem you don't really want to take a decision: you just
write quasi/semi/pseudo/pre in front of the name. But new names can be difficult to invent, to sell and to justify: if you decide to give the name jungle to a proposed prototype of tropical variety, because it sounds to you that in the tropics are plenty of jungles, it is a loose justification and probably will have no future (unless you are Grothendieck).

My question is: Which do you think is the best option?

In fact, the situation can be worse in some cases: what happens if
some name has already been used but you don't agree with the
choice? Is it adequate to modify it, or can it be seen as some sort of
offense?

I could put some very concrete examples, even papers where they
introduce red herrings, new meanings for old names, new names and
no-names for some objects, all in the same paper. But my point is
not to criticize what others did but to decide what to do.

Let me mention as a counterpoint that there is less need for
new terminology than one might expect. Mathematical exposition
is often more successful and clearer without new terminology, and
one should consider whether one needs any new terminology at all.

It seems to be a typical beginner's mistake to name everything in
sight, introducing all kinds of fancy names and cluttering one's
writing with unnecessary terminology and jargon. To be sure, this naming process is easy, as well as fun; one feels like Adam or Eve in the garden. I've succumbed to the attraction of it myself. But now I view this more negatively, for it imposes a
kind of tax on the reader. One opens the article and finds a
theorem stated there:

Theorem. Every big-topped parade is heartily divisible.

The jargon prevents it from having an immediate meaning, even for an expert in the subject, and one must
hunt down the definitions of the various terms. I am sure that many mathematicians have had this experience. Articles are almost
never read front to back, and so the definitions of new terms are often missed. The question of whether the article
will be read at all is often settled by browsing through it and
seeing if the theorems are interesting. The jargon tax is a tax
many readers are not willing to pay — when one can't find
meaningful mathematics easily enough, then one simply looks elsewhere, and one article (perhaps your article!) is dropped in favor of another.

For this reason, I find it desirable, when possible, for one to state one's
theorems in a manner that can be readily understood with ordinary
terminology, even if some new-fangled jargon would make it slightly shorter or would perfectly express some extremely abstract connection.

In time, of course, new objects and ideas find an established usage, and there will be a need for new names. My comment is merely a caution against over-exuberant naming.

Let me address the question "what happens if some name it has already been used but you don't agree with the choice?", by giving a recent example from (mathematical) physics. The 2012 experiment that discovered a "Majorana fermion" in a superconductor attracted much attention because it would be a realisation of a non-Abelian anyon. The name was a red herring, because a fermion by definition has Abelian statistics.

In a discussion on Wikipedia it was argued we could keep the name, for the same reason that "We should not rename the "jellyfish" Wikipedia article into "Cnidaria" just because jellyfish are not fish." But the oxymoron of a "non-Abelian fermion" was sufficiently unpleasant that the name has been banned in favor of "Majorana zero-mode" --- less pretty, without the neat pointer to the Majorana-Fermi duo, but more accurate. Referees played a decisive role in pushing this change through, these days you just can't get a paper published on "non-Abelian fermionic statistics".

So yes, I do think it is appropriate to avoid red herrings in the nomenclature, and if they exist, to modify them, preferrably by a minor change in the name. The change in this recent example, from "fermion" to "zero-mode", was major. A minor change that I recall from longer ago is from "quantum chaos" to quantum chaology.

For me, I think naming is not as important as understanding what it is. We all understand Fourier transform, Banach space, Peter-Weyl theorem, Pauli matrix. If we rename just one of them, what would we call?

As someone who used to work as a translator and is very inquiring about how an idea forms and evolves, I have some chances to observe how a same concept is translated to different terms, and how a same object is renamed several times in different contexts. Perhaps this practice is exotic in academical mathematics, and I admit that I have zero experience in graduate research, but cognitively I don't think our brains are that different.

When does a perfect name come? When you are in a rush. When you are in a rush, you don't have time to think about correctness, you just want to make it quick to solve another important, urgent problem. Your brain will cut all unnecessary information about the object, leaving just enough bit so you can jump to conclusion, or in this case, a name. Those unnecessary bits may be essential to for the concept to form at the first place, but unfortunately, don't really relate to the surrounding concepts in the sentence. The surviving bits that constitute the new name are the ones closer to the surroundings. So even when the original name is short and accurate, it will be replaced by a new name that fits the context.¹

In research, you are expected to be accurate, and you have plenty of time to learn it, but the principle is the same. My advice is to try using the concept to study many more other concepts, and see how your mind reacts with it when you are in a rush. You can also discuss those other concepts with your colleagues, and see how they complete this sentence when you are stuck: "you mean the ______?"

Of course in the realm of cognitive science and linguistic, sometimes you have to accept a sticking bad name. But lucky for us, this is also the realm of math, and unlike jellyfish or pineapple, non-abelian fermion or group without inverses aren't that imaginative, so they will always invoke an unpleasant feeling when reading it. Hopefully one day we can go around them, and let them rest in peace.

Here is a challenge, inspired from the small-world network: try explaining a topic by introducing only 6 intermediate terms. This will force you to twist what you already know about it, so that you can view it in a different perspective. You have to be bold to cut off the details that rock your soul, but by then the big picture will emerge. Only after seeing the big picture that naming can become a piece of cake.

I have an article for this, you can check it out: Making concrete analogies and big pictures.

<sup>¹ This is my own theory, but is inspired from Gentner's Structure Mapping Theory

Related: Hyphens after the prefixes “non-” and “anti-” in mathematics</sup>