For an assignment, I'm looking for examples of times in history where the ownership of specific artefacts have been debated.
For example, someone discovering an artefact, and then having it being taken away for the purpose of study/documentation, or even just presentation, in perhaps a museum.
There's the case involving St. Columba, who borrowed a manuscript and surreptitiously copied it. The lender of the original disputed ownership of the copy. At the time it wasn't an artifact, so it's more an early example of copyright law.
Written in Latin. The Cathach is the oldest extant Irish manuscript of the Psalter and the earliest example of Irish writing. It contains a Vulgate version of Psalms XXX (10) to CV (13) with an interpretative rubric or heading before each psalm. It is traditionally ascribed to St Columba as the copy, made at night in haste by a miraculous light, of a Psalter lent to Columba by St Finnian. A dispute arose about the ownership of the copy and King Diarmait Mac Cerbhaill gave the judgement ‘To every cow belongs her calf, therefore to every book belongs its copy’. The arbitration failed and the Psalter of St Columba passed into the hands of the O'Donnells after the battle of Cul Dremhne in A.D. 561. St Columba went to Iona in A.D. 563. It is possible to date the manuscript late 6th or early 7th century from the script, but modern historical scholarship has cast doubts on the authorship by St Columba.
ref: Royal Irish Academy: THE CATHACH / The Psalter of St Columba
In Indonesia, the recent dispute about historical artifact is concerning Sangguran inscription, popularly known as "the Minto stone", after a British colonial administrator of India and Indonesia. The artifact is a stone inscription dating back to Hindu Mataram kingdom of Java (ca. 7th-11th century). It is estimated to be made ca. 928 AD.
The artifact was unearthed by the British colonial administration, soon after the British captured the French-aligned Dutch East Indies during Napoleonic war. It is a quite important inscription detailing how the Javanese king granted a village autonomy and gifts as reward for their lord's contribution in campaign against Bali. It gives insight on the governmental structure of Mataram kingdom and the religious life there. Coincidentally the inscription dated to the very last year of Central Java phase of Mataram kingdom. The next year after the inscription was made, Merapi volcano erupted and the kingdom was moved into East Java. Since that day, all Hindu kingdoms in Java were centered in East Java (Kediri, Singhasari and ultimately Majapahit)
The artifact is now tragically ended up as garden decoration in Scotland. Its ownership is now disputed between the heirs of Lord Minto, who was gifted that stone by Raffles, and Indonesia. The attempt to repatriate the stone is currently major news in Indonesia.
Kennewick Man or The Ancient One is perhaps a good example, though I hesitate to call human remains an artifact. The argument is also less over ownership in a property sense and more over intellectual/cultural ownership. The primary issue is how much scientists have an obligation or right to study important remains in contrast to the rights of indigenous groups to control the remains of their ancestors. This has a moral dimension, but also a legal dimension because of the passage of the Native American Graves Protection and Repatriation (NAGPRA) act in 1990 which makes some requirements of the government to return human remains and sacred objects to indigenous groups. This is where the legal battle really begins.
These two articles are a good place to start, but there is a lot of information on this case out there. The Smithsonian article is a pretty good overview of the situation, but it has a bit of bias towards the side of the scientists on one side of the legal battle. The second article has less background but covers a lot of the issues from an indigenous perspective. I've also included the Society for American Archaeology (SAA) opinion on the court case surrounding the subject as another perspective that sums up some of the trade-offs in the situation.