Westron evolved from the Númenorean language Adûnaic, which in turn evolved from the ancient language of the Atani in the First Age. "Adûnaic" is clearly, in fact, Adûnaic for "Westron" (!!!)
The clearest statement of this is given in Lord of the Rings Appendix F:
In the years of their power the Númenoreans had maintained many forts and havens upon the western coasts of Middle-earth for the help of their ships; and one of the chief of these was at Pelargir near the Mouths of Anduin. There Adûnaic was spoken, and mingled with many words of the languages of lesser men it became a Common Speech that spread thence along the coasts among all that had dealings with Westernesse.
After the Downfall of Númenor ..... They used therefore the Common Speech in their dealing with other folk and in the government of their wide realms; but they enlarged the language and enriched it with many words drawn from Elven-tongues.
In the days of the Númenorean kings this ennobled Westron speech spread far and wide, even among their enemies; and it became used more and more by the Dúnedain themselves, so that at the time of the War of the Ring the Elven-tongue was known to only a small part of the peoples of Gondor...
Regarding it's even earlier origins, Appendix F again provides the proof:
It was in origin the language of those whom the Eldar called the Atani or Edain, 'Fathers of Men', being especially the people of the Three Houses of the Elf-friends who came west into Beleriand in the First Age...
So the language that became Westron was therefore originally spoken by the Men of the Houses of Beor, Hador and Haleth, was taken to Númenor where it became Adûnaic, and eventually brought back to Middle-earth during the years of Númenorean power where it further evolved into the Westron of the time of Lord of the Rings.
