I think a dual sovereignty role may be what's needed.
The Malvinas have a sparse population; the islands are mostly uninhabited. Why not let Argentina have the uninhabited portions of the islands? They can build new cities, infrastructure, and industries. Right now, the islands are mostly neglected because they are so far away from the UK and they are politically sensitive with Latin America supporting Argentina and the UK's militarization of the South Atlantic causing issues. Argentina, being the closest country and having a lot of pride over having the Malvinas returned, would likely develop the islands to their full potential. This has benefits to the most amount of people. The earth has limited resources, and this seems like an efficient way to use those resources. Argentina can also have all of the natural resource rights around the islands.
I'm sure more solutions can come out if the parties have bilateral talks on sovereignty.
Ultimately, this is something that needs to be addressed by the world community, and should have been addressed following or during WW2, where the British Empire should have been fully dismantled. However, it was ignored, and now we're dealing with it today, and will deal with it in the future in regards to the Malvinas and other territories. On one end, we can dismantle the entire Empire as well as the UK itself. The other would be to preserve the state of the Empire. The result will lie in the middle.