Okay--I'm no halachic expert, but the first brief I ever wrote in law school was on more or less a very similar topic.
According to the Heart Balm Act of 1935 (not sure if this was a Pennsylvania only law, but I know it applies in New York too) a gift given "in contemplation of marriage" only vests when the marriage is complete. So, the bling-bling was NOT rose's from the "engagement", as it never turned into a marriage. (The same would hold true for an engagement ring given, where the fiance, say, cheated on his fiancee and then asked for the ring back. Legally, she'd have to return it)
Now, 1935 is well after the Titanic's unfortunate sinking, and case law from before the Heart Balm Act, shows that it was on a case-by-case basis. Doesn't seem like Rose would get to keep the diamond since she broke the engagement.
From a jurisdictional standpoint, the case occurred in international waters, so it would sort of depend where the Titanic was registered, to determine which law applies.
When Cal gave the coat to Rose, he did not intend to give her the diamond. And although we often say possession is 9/10 of the law, where intent is lacking, it seems likely that the diamond would belong to Cal.
Since Cal cashed in his insurance policy on the diamond, the diamond seems that it would belong to the insurance company, though one might argue that the premiums Jack paid over the years were the cost the insurance company received in exchange for the premium payout.
A very weighty legal conundrum indeed. Finders keepers, losers weepers.
So, what we have so far is that it would seem that halachically there is a strong case that the diamond belongs to Rose, but according to Pennsylvania/New York law it would belong to Cal.
I'm wondering how much halachic weight is attributed to Cal's having "lost hope" of finding the diamond. First, because from Rose's perspective . . . well, she knows exactly to whom it belongs. Furthermore, it's a very distinctive diamond; any random person could have found out who its owner is without too much research. But Rose for sure!
My question is: If I were to find an object that I knew for a fact belonged to a specific person -- I wouldn't have to do any research to find out to whom it belongs, because I already know it for certain-- and I also know for a fact that that person has lost hope of finding it, do I have a halachic obligation to return it?
Ethically I would imagine that I do.