At the moment nobody knows, but the New York Times reports that a judge in Fort Worth is stuck deciding this question.
Oswald was exhumed in 1981 to make sure it was really him in there, and after tests confirmed his identity (or did they?) he was reburied in a new coffin. That's because the original pine coffin was not doing too well after almost 20 years in the ground, as you might have expected. So what does one do with a used coffin? Well, if it's Lee Harvey Oswald's coffin and you're the Baumgardner Funeral Home, you apparently stick it in a closet for another 30 years and then put it up for auction. (The Times has a picture of it sitting in an office near two workers' desks, and that must have been a very pleasant day for them.)
This super-neat collectible sold in 2010 for $87,468, but Oswald's brother Robert sued to block the sale. This week a Texas judge heard arguments in the case and is expected to rule later this month.
There is no question that Robert Oswald owned the coffin for some period of time after he bought it in 1963. According to the Times, the funeral home's argument is that he then effectively donated it to Lee Harvey's estate, and that the heirs to said estate, Oswald's widow (who is still alive) and his two daughters, have never claimed it. I have not yet seen the trial briefs—which I am currently trying to get—but presumably the argument is that after 20 or 25 or 30 years or whatever, the heirs should be considered to have abandoned their claim. Why the funeral home would then be entitled to it, unless it's just by virtue of possession, is not yet clear to me.
Why anyone would want it is also not yet clear to me. I suppose it would make quite a conversation piece if you turned it into a coffee table or something like that. The conversation would probably involve the phrase "what is wrong with you?" but still.
More to come on this creeptastic legal drama.