We can take a look at one of the most famous cabinets of curiosities: that of Ole Worm, in Denmark about the time you're mentioning. Here is a woodcut of the inside of his cabinet. As we can see, the cabinet contains a wide variety of items, from natural phenomena such as taxidermized animals and fossils to artifacts from faraway countries and ancient peoples (mostly from the New World in his case, so a lot of Native American stuff).

In the beginning, the majority of the cabinet would likely have been natural phenomena, such as shells, taxidermied animals, bones, horns, and fossils. The earliest depiction of such a cabinet can be seen here. and we can see that it is heavily reliant on natural phenomena. Later versions of the cabinet might include works of art as well as the aforementioned natural phenomena and artifacts. Here is a later 17th century painting of part of a cabinet which depicts a lot of art as well as shells and ancient coins.

One thing that you might find in a Künstkammer would be a "thunderstone". These were flint points (Paleolithic tools such as arrowheads, spear points, or axes), but people of the time believed they were embedded in the ground by lightning and had magical properties. It was actually the collection of these "thunderstones" and the comparison of them to tools brought to a Künstkammer from the New World that enlightened scholars as to their actual use as tools by very ancient people.