So, ICly the monks will be very interested in the feather but won't want to take it or anything. They'll tell them that it's a good thing to hang onto and that it would be helpful in either location, saying that Ho-oh might react to seeing it or that it might make Lugia more friendly to them.

To let you know OOCly, the prizes are to do with whichever bird characters visit. So it's up to you guys if you want another Lugia item or a Ho-oh item instead. Both birds will react, and the Silver Wing will allow whoever's holding it to get closer to Lugia, while bringing it to Ho-oh might give a clue as to what's going on.