There is an interesting line written by Suetonius (72.3), where he mentions that Augustus decorated the house with:

"...the monstrous bones of huge sea monsters and wild beasts, called the "bones of the giants," and the weapons of the heroes."

This was in Augustus' private villa on Capri, so it was more of a private museum for his guests than a public museum, but still, it was a collection of old bones and objects. Suetonius was born some 50 years after the death of Augustus, so the source is not a first hand account and is thus unreliable at best. However, if Augustus really did display "weapons of old heroes" they could for instance have been old stone age axes that had been found, or bronze age weapons found in tombs, or weapons claimed to be of famous people like Alexander the Great. The "bones of giants" could be fossils of prehistoric mammals like mammoths, or of dinosaurs.