How did the Drúedain come into existence? Were they created by Eru Ilúvatar or one of the Valar?
The Drúedain called themselves Drughu. When the Drúedain settled in Beleriand, the Sindarin Elves adapted this to Drû (plurals Drúin, Drúath) and later added the suffix -adan "man", resulting in the usual Sindarin form Drúadan (plural Drúedain). Tolkien also used the form Drûg, with a regular English plural Drûgs
The Drûgs were the first to migrate from the site where Men awoke, in the east of Middle-earth. A band lived among the Second House of Men, the Haladin, in the First Age in the forest of Brethil, whence the Elves came to know and love them. Although a number of the Drúedain came with the Edain to Númenor, they had left or died out before the Akallabêth, as had the Púkel-men of Dunharrow. At the end of the Third Age the Drûgs still lived in the Drúadan Forest of the White Mountains, and on the long cape of Andrast west of Gondor. The region north of Andrast was still known as Drúwaith Iaur, or "Old Drûg land".
It looks like they were created like the rest of Men, they just made some different choices along the way.
Link to the full article: http://en.wikipedia.org/wiki/Dr%C3%BAedain
I think Tolkien did his best to explain the diversity in human 'breeds'...(his time-frame was a little too short, but nonetheless)...he described the people of Haleth as short, stocky, and secretive...they lived in the forest of Brethil, and were one of the 3 houses of the Edain.
We know that the house of Beor was killed off (with limited, but renowned survivors), and we know that the house of Hador was tall and blonde. The pukel-men were depicted in the same way by those who had seen them long before: squat and secretive, as were the people of Ghan-buri-Ghan (whose language was very different..unlike the common tongue...partially due to their isolation, partially due to its somewhat different roots - the people of Haleth were said to have a language unlike the other Atani).
It is a reasonable guess that when Beleriand was broken, the people of Haleth wandered east, and sought isolated places like the Druadan forest ("Dru-adan - land of drughu-men")
To build on @Deleteman's answer with actual quotes from the books.
We can thus if we wish elaborate the acient legend of the coming of the Edain in The Silmarillion by the addition of the Drúedain, descending out of Ered Lindon into Ossiriand with the Haladin (the Folk of Haleth). Another note says that historians in Gondor believed that the first Men to cross the Anduin were indeed the Drúedain. They came (it was believed) from lands south of Mordor, but before they reached the coasts of Haradwaith they turned north into Ithilien, and eventually finding a way across the Anduin (probably near Cair Andros) settled in the vales of the White Mountains and the wooded lands at their northern feet.
–Christopher Tolkien, Unfinished Tales, Part III: The Third Age, Chapter I: The Drúedain; Further Notes on the Drúedain
This is the closest reference to the Drúedain's origins that you can get. If you believe these "Gondorian historians", then the Drúedain originate from the lands south of Mordor (it wasn't called Mordor then).