I've been a long time reader of this sub even before I had an account so I know you all may know a little about this, as I have no idea and do not trust just simply googling it for a nice answer you all usually provide. Thanks!
The earliest known museum is "Ennigaldi-Nanna's Museum", in Ur in Mesopotamia circa 530BC.
In 1925, archaeologist Leonard Woolley, while excavating a Babylonian palace discovered a room full of artifacts from many different periods and places. Woolley began to suspect that he had found an ancient museum, and his suspicions were confirmed when he found cylindrical clay rods, inscribed in 3 languages, which turned out to be labels for the artifacts in the room.
We know that this museum was established by Princess Ennigaldi, with the support of her father, King Nabonidus (who restored the great ziggurat in Ur).
The contents of the museum included boundary stones, foundation stones, statues, clay tablets, ranging from 700 to 1600 years before the museum was built.
We have no record of why the museum was built.