Having trouble with a duplicate "Red Eagle" quest in Skyrim

Solution 1:

According to the Elder Scrolls Wiki, you can just complete the quest again.

The bugs section of the Red Eagle page says

Once this quest has been completed, it is possible to start it over again by reading the book The Legend of Red Eagle. Some Forsworn NPCs will have the sword in their possession (which implies that there are multiple copies of that sword throughout Skyrim). If you have restarted the quest but do not intend to complete it again, do not take Red Eagle's Fury, as it will count as a quest item again (you won't be able to drop it without using console commands - alternatively you can insert the sword back to its place inside Rebel's Cairn to get rid of it). If you do end up with multiple Red Eagle's Fury swords, you can deposit them in the chest in the main chamber of Rebel's Cairn to get rid of them (PS3 and PC). It is not known if the possibility to restart this quest is intentional or if it is a bug.

So, if you get another Red Eagle's Fury, possibly from another Forsworn, then you can just put that in the spot in the cairn and finish the quest.

If you don't have another Red Eagle's Fury, the Legend of the Red Eagle page's bugs section says you can use your Red Eagle's Bane from the first time you completed the quest:

Placing Red Eagle's Bane in the weapon slot on the pedestal will mark the quest as complete if stuck.