There is not actually one definitive list of epic hero characteristics. Different sources identify different traits, depending on their own perspective. The list in the previous post is pretty complete--I've seen some shorter lists and some longer ones.
The list in the previous post mentions nobility that is gained through battling a powerful foe as one trait. Some sources instead state that an epic hero is born into nobility, usually by being related to a monarch of some sort.
Another quality often attributed to the epic hero is humility, although there are at least two very famous examples of epic heroes who do not possess this trait. Beowulf can barely open his mouth without boasting of his own prowess and accomplishments. Odysseus also has a prideful streak that sometimes got him into trouble.
The point isn't that a character possess every possible trait of the epic hero. If they are powerful in some way and reflect the values of their culture while battling fearsome beings, then they are probably an epic hero.
There are typically nine characteristics associated with an epic hero. The following characteristics are those which an epic hero possesses.
1. The epic hero is introduced to the reader (or listener (if the tale is told in a traditional way (through song or verbal story telling)) in the midst of turmoil--in media res.
2. The epic hero is a leader, warrior, and a polished speaker.
3. Often, the epic hero is a demi-god and/or possesses a weapon bestowed upon him (given the epic hero is typically male) by a god. The weapon is large, can be well-known, and/or an heirloom.
4. The epic hero's journey must test his courage and skill. Many times, this journey includes descent into the Underworld.
5. The epic hero will always take on challenge/s that no others will (regardless of the anticiapted outcome).
6. The epic hero must either possess or gain aristeia (nobility)-- through battling a foe only equal or greater then himself.
7. The idea of arete ("bringing virtue to perfection") appears.
8. Whatever qualities or abilities the cultural of the epic hero raises up, the epic hero has an abundance of.
9. The antagonist is normally a God/god hater and the climax happens when the protagonist and antagonist meet for their epic battle. The climax is delayed in order to build suspense.