I've found an essay on the subject from a source that is naturally biased towards concluding that he's a Jew, but ultimately rejects the thesis:
https://www.counter-currents.com/2012/11/was-stalin-jewish-and-does-it-matter/

Most of the arguments for his Jewishness are onomastic, pointing to his and his close ones' names as evidence. However, a closer look at etymologies show those claims are very weak.

>Turning to Mr. Tarrell’s article as a cogent point of reference, his first contention is that Stalin, born Iosif Vissarionovich Dzugashvili, had been given a name that would have been unthinkable for a non-Jew in Czarist Russia: Iosip (Joseph).[4] The question arises, however, as to why the christening of Gentiles with Old Testament names in Czarist Russian would be “unthinkable,” since the Orthodox Russian tradition is also drawn from the Old Testament? Rather, Iosip is a variation of Iosif, which is the Russian, Romanian, and Greek form of Joseph.[5] Another Russian variation is Osip. The “p” in Iosip can be found in various European languages. The Hebrew version is rendered as Yosef.[6]

>Terrall states of Stalin’s given middle name, Vissarionovich, that “there is evidence that the name Vissarion is a Jewish name. For example, Vissarion Bielinsky was a Jewish writer in Russia…”[11] Vissarion is derived from Greek, Bessarion, and means “who he gives new life.” His father’s name was spelled Besarion.

>The primary case for Stalin’s alleged Jewish origins is that his name Dzugashvili (variant: Jughashvili) means “son of a Jew” in Georgian, Dzu supposedly meaning “Jew.” Terrall states a relatively widely believed claim that “Dzu means Jew.” However, according to Montefieore, the name means “son of Juga.” Montefiroe states that the name is derived either from Ossetian, meaning “herd” or Georgian (djuga) meaning “steel.”[12] The latter Georgian meaning would surely account for the adoption of “Stalin,” “Man of Steel,” by Iosif Vissarionovich.

>Stalin’s great grandson, Jacob Jugashvili, writes:

>There is no word Jew for Jews in the Georgian language. . . . Jew in Georgian is Ebraeli, so the theory of “son of a Jew” (which is very tempting considering our name in light of its English spelling) is simply, wrong.[13]

>Of this “Jew” sounding part of the name, Jacob Jugashvili explains that usually the surname is written is Russian as Dzhugashvili because there is no “j” in the Russian alphabet, so Russian uses the letters “d” and “zh.”[14]