So the current rumor in our RTL campaign is this:

Split Heirlooms

and the reward is:

The heroes choose one weapon in their possession and attach this card to it. That weapon gains +1 damage and cannot be destroyed.

Between the wording of choosing a weapon in their possession and the rulebook stating:

The Rumor level is played according to the Quest Guide
description and rules, which also specify the conditions the
heroes must meet to receive the Rumor reward. The reward is
granted immediately when the conditions are met. The Rumor
card is also then removed from the campaign (being placed in
the graveyard box), unless the card instructs the heroes to keep
it. Whether removed from the game or kept by the heroes,
however, that Rumor card is no longer considered to be the
active Rumor.

I believe that this means they must choose one of their current possessions . If we happen to still be at the copper level of the campaign then they are stuck with an unbreakable copper weapon.

A couple of my players disagree and say that they get to choose when to apply the reward to the item of their choice.

I told them I would put the question to the forum and see what opinions present themselves.

Thanks

The rumor reward says an item in their possession and attach this card to it. Since the rumor is rewarded the instant the its conditions are met, I think that most certainly means that it is attached to an item they have at that time. The immediacy is at the very least strongly implied by the card and the wording in the rulebook.

There is a rumor reward that can be kept for later, which allows the hero party to teleport anywhere on the map (a very good rumor btw: it was very helpful in our campaign), but it explicitely says that you can keep it for later. Here it is not the case, and the way it is written, I don't really see how your players can argue that they can keep it for later.

There are two copper weapons that are actually worth making indestructible: the Staff of the Grave and the Falcon's Claws. The first is a good magic weapon to have even in the Gold age (undying is so annoying!), and the second is dangerous for Lieutenants.

In addition, the heroes always have the choice to pick a new rumor to replace and save this one for later campaign levels.

Parathion said:

+1.

The heroes are allowed to read the rumor card before accepting it. If they don't want this reward at Copper level then they should draw a new one and leave this card in the deck for later. If I'm not mistaken, they can also replace this rumor even after they've accepted it, so long as they do so before completing the rumor dungeon. So even if they change their mind later, they aren't stuck with it.

The only way this question could become a problem is if the heroes don't stop to think about it until they've completed the rumor dungeon and are applying the reward. That's a lack of foresight on the heroes' part, not a flaw with the game. The Advanced Campaign requires both Heroes and OL to focus on the big picture if they want to come out ahead. If you treat it like it's just a series of dungeons leading up to a final conflict, you're missing out on the purpose and you'll probably end up screwing yourself somehow.