The Iliad and Odyssey definitely fit the bill here. Now, we tend to think of them as great written works of Greek literature, but their origins predate Greek writing. The prevailing theory (for the Iliad, at least) is that they are derived from multiple oral traditions, which would have been short enough to perform in one night. There are also other indicators that show they are originally oral works. There is a lot of restatement, which would have helped with memory (for example, there are several scenes where someone says something to a messenger, and then it is reported in the exact same words immediately after), as well as type-scenes (many battles are described using the same language). Finally, there are epic epithets for many of the characters, which fit half a line, making it more formulaic. (e.g. Hector, the breaker of horses; Zeus, the father of gods and men)

Some of the verses are very old, as well. Due to the meter, we can tell that some verses had words that would have originally had the letter digamma, which fell out of use in most Greek dialects by the 7th century BC.

Besides this, you may want take a look at Hesiod's works. They're considered to be younger than the works of Homer, but are a bit more folkloric, giving at-times conflicting accounts of mythological occurrences, and including a section on when you should plant certain crops based on the position of the stars.

Also, it's worth noting that the Iliad and Odyssey were part of a larger body of oral epic known as the Epic Cycle. Unfortunately, the other epics in the Epic Cycle come down to us only in fragmentary form.

Can't say for certain this is the earliest, but as a start, Beowulf. The original surviving manuscript written in Old English dates to somewhere around the 8th-11th centuries, but it's possible (if not probable) that the story had been carried down for hundreds of years as an oral tradition (similar to that of the Illiad & the Odyssey). The events of the story take place somewhere around the 5th century AD.

Speaking of the Illiad and the Odyssey, if you're willing to consider them extremely early European (which I imagine most historians would), then they'd be far older than Beowulf. It is estimated that these works were first written around the 8th century BC (~1300 years earlier than Beowulf). However, the events depicted seem to have happened at or around the Late Bronze Age collapse, approximately the 12 century BC.

To emphasize how seminal these works were in early European consciousness, the first Roman Emperor, Augustus, commissioned one of the great Roman poets Vergil to write a segue called the Aeneid. It picked up the story from the end of the Illiad, with the hero Aeneas fleeing the ruins of Troy with his family, embarking on an epic journey and traveling all they way from modern-day Turkey to modern-day Tunisia. Finally, they make their way to the Italian peninsula, to the area around modern-day Rome. The work was a bit of clever propaganda on the behalf of Augustus Caesar to provide legitimacy for his father Julius Caesar's rule, and thus also Augustus'. Aeneas, by the way, was the son of the goddess Venus (greek: Aphrodite), so by tying his bloodline to Aeneas, Augustus Caesar was explicitly claiming his divine bloodline.

It's worth mentioning Irish mythology here. Like other examples mentioned in this thread, much of it had old roots, but wasn't captured in written verse or prose until we entered the Christian era, and what we do have is based on old manuscripts that were copies of copies.

One item of interest might be the Book of Invasions (in Irish, Lebor Gabála Érenn, literally 'The Book of the Takings of Ireland'). It's fairly typical for Irish myth: the versions we have are based on documents dating back to the 12th century and later, which themselves were based on now-lost manuscripts of at least a few centuries earlier. Here's a translation from the late 1930s.

That translation is based on the version found in the mid-12th century Book of Leinster. This also contained The Cattle Raid of Cooley (in Irish, Táin Bó Cúalnge; translation), a fun epic tale of warfare between the ruling clans of Ulster and Connacht, featuring the great warrior Cuchulainn.

Lastly, it's also worth considering Irish place-name lore ('dindshenchas'). There's quite a bit of mythology around the names of places, and much of this has been written down in the same old manuscripts. Unfortunately, I'm not as familiar with these, but maybe someone else can say more.

For lay-person intros to some of this old mythology, I might suggest Over Nine Waves by Heaney, or Early Irish Myths and Sagas by Gantz.

Hope that helps!

EDIT: I should mention too, the Cattle Raid of Cooley is part of the Ulster Cycle, which allegedly takes place roughly around the time of Christ (before Christianity came to Ireland of course). The timing of the tales in the Book of Invasions would pre-date this, since those are basically stories of successive waves of migration involving the "gods" and, eventually, the Gaels.

EDIT 2: Changed the book links from a certain online bookseller Leviathan to Worldcat. Sorry, forgot the linking rules for a minute.

EDIT 3: Thanks for the gold!!

Three of the oldest bits of European folklore to manifest in early sources are: 1. the story of the incredible journey and magic flight from the monster that manifests in the Ancient Greek tale of Jason and the Argonauts; 2. the story of the ogre who is burned by the hero; this manifests in the Odyssey with the Cyclopes who is told his assailant's name is "Nobody" and so he says "Nobody has hurt me." In Northern Europe this often manifests as "Self" - and a voice in the forest responds "well if you hurt yourself, you only have yourself to blame."; and 3. The "The Two Brothers" which appears in Ancient Egyptian sources but is also common in Europe.

Each of these stories have ancient manifestations but they are also widespread in Western Europe, attesting to their age and longevity.

edited to replace "Twin Brothers" with "The Two Brothers."

Hi folks,
There are a lot of great comments in this thread, but we've had to remove a number that just name a particularly old piece of folklore or are solely a link to Wikipedia. In regards to the first, simply name-dropping an old legend doesn't really answer OP's question, who is looking for the oldest indigenous folklore, and expects a great deal more of information than a simple one-sentence answer. As for the Wikipedia links, those run afoul of our long-established rule against answers consisting simply of a link.