Multiple copies of artifacts or minerals

Duplicate artifacts and minerals can simply be sold or put into your delivery box to make extra money. As mentioned in the comments, the minerals may also be used as gifts for villagers or as materials for crafting. A limited selection of artifacts are also good gifts for select villagers. Aside from these uses, they serve no other purposes once you've donated one to the museum.

To give you some examples, Abigail loves Amethysts, the Dwarf likes to receive dwarven artifacts as gifts (scrolls, helmet), and Earth Crystals are used in the recipe for Mayonnaise Machines. If you prefer to sell these items instead, artifacts and minerals are all worth a decent amount of gold (except for dwarven scrolls), and are a nice way to have some extra income.