Multiple copies of artifacts or minerals

Besides some of the crystals, I am having a hard time finding a use for my multiple copies of artifacts or other types of minerals.

Will I ever find a use for my left over artifacts or minerals after I donated the first one to the Museum?

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.

You can ship them for extra money, or give them to people as Gifts. I know that the Dwarven Scrolls only sell for 1G but make decent gifts for the Dwarf, as one example.