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.
Best Answer
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.