The are some parts of math that you encounter easily new structures,
obtained by modifying or generalizing existing ones. Recent examples
can be [tropical geometry][1], or the theory around [the field with one element][2]. If one works in that areas, one cannot avoid the problem to name 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][3]
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 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 doesn't pretend to have an 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 will have to put a
name to it (may be 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 terminology), and they
contain some information which is useful, but sometimes 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 call a **jungle** a
proposed prototype of *tropical variety*, because it sounds to you
that the tropics are plenty of jungles, it is a loose justification and
probably will have no future (unless you are Grothendieck).

My question is: What do you think it is the best option?

In fact, the situation can be worst in some case: what happens if
some name it 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.


  [1]: http://en.wikipedia.org/wiki/Tropical_geometry
  [2]: http://en.wikipedia.org/wiki/Field_with_one_element
  [3]: http://ncatlab.org/nlab/show/red+herring+principle