It bears noting that for about the first decade, the term usually appeared in lower-case and scare-quotes as "common tongue" -- that is, not the proper name for any specific language. For example, the very first appearance in original Dungeons & Dragons (1974, Volume 1: Men & Magic, p. 12) says this:
LANGUAGES: The "common tongue" spoken throughout the "continent" is known by most humans. All other creatures and monsters which can speak have their own language, although some (20%) also know the common one. Law, Chaos and Neutrality also have common languages spoken by each respectively. One can attempt to communicate through the common tongue, language particular to a creature class, or one of the divisional languages (law, etc.). While not understanding the language, creatures who speak a divisional tongue will recognize a hostile one and attack.
As usual for OD&D, that's incredibly sketchy -- it broadly assumes that the reader is familiar with the fantasy literature corpus at the time and filling in details appropriately, e.g., as by Tolkien or others. That paragraph is the entirety of the discussion of languages in the original three volumes.
Even in the Advanced Dungeons & Dragons rules (1977-1979), the books are still writing the phrase in lower-case and commonly in scare-quotes. The Player's Handbook states rather clearly that it does not indicate any particular or universal language, even within a single game-world (p. 34):
All humans, as well as those semi-humans and non-humans in close contact with people, speak the "common tongue". This language is spoken by all states in the central campaign area, but your referee may well have areas in which the common tongue is different from that which your character speaks.
Interestingly, the earliest proper-noun capitalization I can find is in Dragon Magazine #1 (June 1976) within a guest article on languages by Lee Gold (of Alarums & Excursions fame). The earliest official D&D publication is the Holmes Basic D&D book (1977, "Elves speak... Common speech" etc., p. 6), although elsewhere it still uses the original "common tongue" in quotes (p. 9). The Moldvay Basic D&D (1981) is the first that almost always refers to it as "Common" with a capitalized, single-word term (p. B13):
The "common tongue" (or more simply "Common") is spoken by most humans, dwarves, elves, and halflings.
It also appears as the capitalized name of a particular language in Gygax's World of Greyhawk folio (1980). To me, it appears like Advanced Dungeons & Dragons 2nd Edition is the first to universally use the single-word term, to avoid introducing it in scare-quotes, and to say somewhere that it's maybe some kind of global language (1989, Player's Handbook Glossary):
Common - the language that all player characters in the AD&D game world speak. Other languages may require the use of proficiency slots.
However, 2nd Edition in the broader text still talks about regional languages as being the basic starting assumption (e.g., PHB p. 20): "Human PCs generally start the game knowing only their regional language - the language they grew up speaking"; and all languages are spelled lower-case. 3rd Edition is the first to commit to using capitalized keywords as a device, and "Common" is used in that context without exception, seemingly as the in-world name for a particular language.
In summary: The earliest editions either implied or outright stated that "common tongue" was a placeholder for some regional in-world dialect, not a proper name for any particular language, and certainly not a universal or even global one. My reading is that the original author surely had in mind Tolkien's Westron, which he also calls "Common Speech", as a canonical example. Later editions evolved in the direction of shaving off this inherited linguistic nuance.