So after reading the rules and FAQ, I think I know the answer to this, but I wanted to check.

I have Core Set Eddard Stark (Renown. Deadly. Stalwart. Eddard Stark claims 1 power when he comes into play.) out, and toss a duplicate on him. Later, to save Eddard (for whatever reason), I discard the duplicate. Now, since duplicates have no title, text, or traits, the duplicate doesn't have Stalwart on it. Once it hits the discard pile, however, it is no longer a duplicate, but back to being Ned, with Stalwart.

So here's the question. Would the duplicate's Stalwart trigger, hence putting the Ned back on top of the deck? My hunch is no, because from what I can tell of the timing, the card would need to be Stalwart before it hits the discard pile.

Correct. Stalwart doesn't work from the discard pile. The dupe doesn't get its text back until it is in the discard pile. A dupe of a Stalwart character ends up in the discard pile when it is discarded.