An instance of the SET-PACKING problem is given by a list of sets $\mathcal{S} = \{S_1,\dots,S_m\} \subseteq 2^U$. It is a ``yes'' instance iff there exists some subset $\mathcal S'$ of $\mathcal S$ of size $\ge k$ such that for every $S,T \in \mathcal S'$, $S\cap T = \emptyset$.
Now, I propose the "promise variant" of this problem: I am telling you that your instance is a yes instance; that is, there exists some $\mathcal T'\subseteq \mathcal S$ such that all elements of $\mathcal T'$ are disjoint and it is of size $\ge k$. Can you find some $\mathcal S'$ that is of size $\ge k$ with all disjoint elements?
This seems to be an obviously NP-hard problem, but one needs to be slightly careful in the proof: intuitively, simply taking an instance of SET-PACKING and appending k arbitrary disjoint sets will work (informally, if finding them is in P, then SET-PACKING is also in P).
This relates to a similar question that appeared in cs.stackexchange. I am aware that promise variants of NP hard problems are generally hard (that is, given an NP hard problem X, define promise-X to be the problem X when a yes instance is guaranteed).