I'm a merchant in the Byzantine empire in the 11th century, who finds a hoard of old Roman coins from the 1st to 3rd century. Would these coins still be considered monetized in the empire? Or would they only be worth their melt value/ value as a curiosity?

This winds up being a complicated question.

Although the chain of government was solid between the time of say, Trajan, to the 11th century, the monetary system was not. The aureus of the 1st and 2nd century was a gold coin of high purity. Its weight declined somewhat through the period, but contained around 7g of gold.

The gold coinage of the Eastern empire was actually in fluctuation during the 11th century that you mentioned. The gold coin at the beginning of the 11th century was the solidus, which was a coin of high purity containing around 4.5g of gold. A financial crisis resulted in a series of emperors debasing the solidus several times throughout the 11th century, to the point where the solidus had very little gold at all by the beginning of the reign of Alexius I. Alexius would eventually discontinue the debased solidus altogether, and replace it with a new, highly pure gold coin called the hyperpyron.

Throughout the Roman period, mint officials would engage in a practice called countermarking. If you had an old or possibly heavily worn coin, it could be weighed, and if the weight was up to standard, it would be given an official mark that made it eligible to pay taxes with, and thus desirable for circulation. So your merchant could easily have his antique aureii countermarked.

But he wouldn't have even thought about doing that. In coinage there is a principle called Gresham's Law, which states that "bad money drives out good". If two forms of money have an equivalent face value, the one with the higher intrinsic value is retained by people while the lower value form circulates. My grandparents kept jars of silver dimes hidden in a closet after the US coins lost their silver. The ancients did this too. When Caracalla released the debased antoninianus to replace the old silver denarius, denarii were hoarded.

In this case, the merchant is holding coins in his hand that contain 7g (or more) of actual gold. The solidi that the mint is releasing have, at best, 4.5g of gold and maybe a lot less if we are at the end of the 11th century. That coin hoard is going to be worth way more in melt value than anything the mint official will value them as.

Sourced, in part, David Vagi, Coinage and History of the Roman Empire