I was just at a conference where someone asked me if I had coined any mathematical terms. Well, sort of! I was the one who decided on the name “FI-modules” for the abelian category Tom Church, Benson Farb and I wrote about in this paper. More informally, I’m pretty sure I’m the originator of using “Bhargavology” to mean “the program of counting arithmetic things by putting them in bijection with orbits of the integral points of a group acting on the integral points of a space.” At least, I can find this usage in emails I wrote in 2003, after Manjul’s thesis but before any of the papers came out. And that still seems to be something people say.
My coinages have not always been successful. Nobody ever again mentioned the “esperantist graphs” from my paper with Hall and Kowalski. (They were named so in honor of Harald Helfgott, who speaks Esperanto, and because in some sense they are typically graphs we hope are expanders.) Nor did “superduperstrong approximation” catch on.
Quomodocumque 
I coined the term “Tring group” for something related to the String group as the orthogonal group O(n) is related to the special orthogonal group SO(n). It’s not my area, but at a conference talk in 2002 I heard of this thing and shouted out from the back row of the auditorium “You should call it the Tring group!”
The analogous coinage “Pin(n)” for the thing related in that way to Spin(n) was attributed to Serre some sixty years earlier in a footnote to the paper by Atiyah-Bott-Shapiro.
Of course it’s possible that I am not the first one to suggest “Tring”, but I believe there is reason to think that I
oops, meant to say that there’s reason to think that my interruption made a difference
Moi? I’ve got a million of ’em … but my faves are these two acronyms —
☞ Riffs & Rotes
Huh, I thought of this as something that happened all the time. Just thinking of terms I’ve definitely seen other people publish using since: honeycombs, hives, puzzles, subword complexes, branchvarieties, positroid varieties (didn’t really invent the concept but connected the concepts), Bruhat atlases.
Sylvester was really into such creations, e.g. “matrix” and many others you know, but also many you probably don’t, like “catalecticant”.