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–x2” 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, Uber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift, 8 (1920), pp. 171–181.
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