Flintlock will gladly accept new historical exhibits in his museum, which you can run into during the exploration of the world. This mainly concerns the items connected with the first installment of the game and all sorts of Easter Eggs. Often, it is possible to find them in graves and in toasters. In general, it pays off to keep all white items and visit Flintlock and Mercaptain on a regular basis to find out if they are interested in any. In the descriptions of some of the items, it reads that there is someone interested in obtaining one, at the Citadel" but the majority of them will not be provided with such a note. For each of the exhibits, you will be rewarded either with an item, with money or with experience points. Also, you will receive a requisition form that entitles you to a free piece of equipment offered by Mercaptain. The quest will be deemed completed after you bring back the first exhibit, but you can keep bringing them anyways.
The list of the exhibits that you can bring has been provided in the unique items chapter.