If there's one wrapped gift per person, you could do it as a white elephant exchange with pick position determined by a game of Settlers. Or Dominion, or Agricola, or anything else that produces a score that can be used to determine not just who won but 2nd place, 3rd place, etc.
Whoever got the low score opens a present. It's theirs! For now.
Now, everybody takes turns choosing a present, starting with the second-lowest scorer and working up to the winner. When it's your turn to choose, you may either open a new present, or steal an opened present. If your present gets stolen, you're presented with the same choice; steal somebody else's, or open a new one. (A given present may only be stolen once per round.) The round lasts until somebody finally grabs a previously unopened present.
Thus, whoever goes last gets the choice between the best of the N-1 known quantities, or take a chance on the Mystery Box Of Mystery.
I do this with both friends and family every Christmas, with the order being determined by random draw rather than a game. Makes for a good time.