What are Central Limit Theorems and why are they called so? I know two opinions:
1) "Central" means "very important" (as it was central problem in probability for many decades), and CLT is a statement about Gaussian limit distribution. If the limit distribution of fluctuations is not Gaussian, we should not call such statement CLT.
2) "Central" comes from "fluctuations around centre (=average)", and any theorem about limit distribution of such fluctuations is called CLT.
Which is correct?
 A: One of my teacher in Probability once told us that this name (Central Limit Theorem) was just used (at the beginning) to stress the importance of the result -which plays a central role in the theory. 
Besides, the ambiguity led to several different translations, corresponding to both interpretations of the term "central". (e.g in French, we can find "théorème central limite" and "théorème de la limite centrale")
A: There's a nice little book on this subject: The Life and Times of the Central Limit Theorem. The book goes over the history of the theorem from its embryonic form to its more or less final form 200 years later.  It also gives precise statements of some of the modern variations of the theorem and indicates directions of research. 
A: From the introduction to History of the Central Limit Theorem: From Laplace to Donsker by Hans Fischer:

The term “central limit theorem” most likely traces back to Georg Pólya. As he
  recapitulated at the beginning of a paper published in 1920, it was “generally known
  that the appearance of the Gaussian probability density $e^{-x^2}$” in a great many situations “can be explained by one and the same limit theorem,” which plays “a central
  role in probability theory” [Pólya 1920, 171]. Laplace had discovered the essentials
  of this fundamental theorem in 1810, and with the designation “central limit theorem
  of probability theory,” which was even emphasized in the paper’s title, Pólya
  gave it the name that has been in general use ever since.

Fischer refers to the paper by G. Pólya, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift,  8 (1920), pp. 171–181.
Edit. The paper is reprinted in George Pólya: Collected Papers, Volume 4, MIT Press, 1984. R.M.Dudley mentions in his comment on the paper that

Although the name "central limit theorem" for the normal limit law seems to have been articulated in the mathematical folklore by 1920, Feller in his famous text attributes to Pólya the first written use of this term.

