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 article by G. Pólya, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift, 8 (1920), pp. 171–181.

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.

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.

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 article by G. Pólya, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift, 8 (1920), pp. 171–181.