If you dispose of any of the items required for the quest prior to turning them in, you can obtain them again by completing the relevant Trial battles again.
If you have turned in an item for the quest but not yet completed it, you are still allowed to choose that item as a reward after defeating the relevant avatar, so choose carefully!
Mamaulabion will inform you of the items delivered thus far, as of the May 2011 update.
Upon completion of this quest, the above items no longer appear in the rewards list for defeating the Prime Avatars.
Mamaulabion (Near the Auction House, Norg)
Mamaulabion would like you to bring him any "thuper-dee-duper, overflowing with thpethial power itemth" you happen to come across in your travels. However, he was not able to provide you with any details on their whereabouts.
SPOILER WARNING: Click here for additional details about this story line.