What is the general protocol to name something after somebody (some people)? In mathematics (and other sciences) there are thousands of concepts, theorems, lemmas, etc which are named after some mathematicians (scientists). However, this nominations are not always very straightforward, especially if we are going to assign a new name for new concepts. For example, I can imagine the following scenarios and I would like to know what general protocol we should apply in each case: 


A concept may have several origins in different fields and due to several individuals. 
A concept is built on another concept which already have a name and this can happen several times. For example "Hecke pairs" is a concept in mathematics, then Bost and Connes make a particular Hecke pair famous, so we have "the Bost-Connes Hecke pairs". Should we name all influential people  in each stage of advancement of a concept?
If an author invents a concept, is it appropriate to name it after him/herself,  or he/she should wait others call it after his/her name?
A concept was invented by some author "X" in long time ago, and then it has evolved to something very modern and somehow different. Should we still call it by the name "X"? 


Please do not hesitate to add new items if you can imagine other scenarios too.

Finally, I would like to ask another related question:

Can we use acronyms in stead of the full names, especially if the names of several people are involved?
Good questions.  I will only tackle the last two:

3) In mathematics, it is (virtually?) universally bad form to name something after yourself.  This is high on the list of things that amateurs/newbies do that make the professionals/veterans roll their eyes.  

Some people have joked that the best strategy to get something named after yourself is to give your nice new concept such a terrible name (or lack of a name) that the rest of the community converges on naming it after you.  

Even after something has been named after you, it is not necessarily completely kosher to speak your own name when referring to the concept.  Rob Kirby famously speaks of "the calculus of framed links" (or, I think ironically, "the calculus") where others speak of "the Kirby calculus".  At one point Armand Borel writes of "the subgroup whose name I have the honor to bear".

It gets a bit ridiculous: when you give a talk and state one of your own theorems, it is most common to write out the names of your coauthors and not write out your own name.  In my student days I saw a lot of first letter then dashes.  Nowadays I mostly see the name dashed out entirely.  Come to think of it, this reminds me of the Jewish practice of leaving letters out of the name "Jehovah", although the theological implications of treating your own name this way are much more profound.

4) I don't know whether we "should", but we often do. In general, it seems to me that mathematics has gotten used to naming things after certain people, and we often name things after people who would never have understood the things that are named after them.  The example Hilbert space (coined by von Neumann in its present generality) is a famous one.  The example Euler system has always struck me as being especially ridiculous (I asked my advisor about this, and he told me that the name comes from Euler products: that's quite a stretch).  

Some people in mathematics are somehow especially good at getting things named after them.  In my field, perhaps the outstanding example is John Tate: he has curves, algebras, half of the Shafarevich-Tate group, half of Hodge-Tate weights, half of Lubin-Tate formal groups, a pairing...As a graduate student, I was struck by the fact that I was giving a talk on Galois cohomology of products of Tate curves, analyzed via Tate local duality.  The title of the talk, "Tate-Tate-Tate Stuff" was a riff on the title of the previous speaker's talk ("Hodge-Tate Stuff") and this Tate-ish ubiquity.  When Tate showed up for the talk, I got very nervous...but he was cool with it.  

Needless to say, John Tate is a true luminary.  The fact that so many things bear his name is only possible because of the immense amount of fundamental work that he did.  But the converse does not hold: e.g. Barry Mazur is a mathematician with a similar impact on the field, but he has...what?  A manifold and a swindle?  (Both of these come from his work in topology at the beginning of his career.)   Instead we have the Eisenstein ideal.  These things are strange.