The most challenging question for me is why Tolkien originally wanted Beren to be an Elf because it is so much a part of my Tolkien landscape that Beren was a Man.  Here are a couple of answers on that front.  One, he was mainly focusing on writing about Elves at that point in his writing history.  He was just more interested in the deeds of Elves, and wanted to keep the Men few and far between.  Two, having Beren be an Elf means we only have one case of an Elf/Man marriage rather than two.  This gives Tuor and Idril more importance, and more significantly, it makes Earendil that much more critical as our only Peredhel.  As for Beren being a Noldo, well, Thingol at least sees this a step down for his half-Maia daughter.  And I have some vague sense (it's been a while since I've read all of BoLT) that in BoLT, the Gnomes/Noldor come off even worse than they do in the later Silmarillion.  Mostly thralls of Morgoth and that sort of thing.  (Thingol's attributing that to Beren even carries through to the published Silmarillion when he insults him at their first meeting.)  That at least gives a dynamic to the Beren/Tinuviel relationship; we need some reason for Thingol to oppose the match!

Now, why make Beren a Man?  The main reason, of course, is to explore the mortality/immortality dynamic that so intrigued Tolkien.  It also gives Thingol even greater reason to oppose the match--a Noldo is bad enough, but some pitiful human?  But I suppose that imagining the ramifications of Beren being an Elf add another reason.  Suppose that the Elf Beren was killed as always, and Luthien gets both of them reincarnated as Elves in Beleriand (so everything the same, except that they are both Elves in both incarnations).  What then of the Silmaril?  We may assume that the conflict of Thingol and the Dwarves happens the same way, and Beren and Luthien get the Silmaril.  Do they then get murdered (say by the Sons of Feanor) so that Dior, then Elwing can inherit the Silmaril?  Because we do need Elwing to get it, to pass on to Earendil.  Or do Beren and Luthien survive and willingly give it to Earendil for some reason?  Difficult questions to explain.

What if they survived to the Third Age, or did a Glorfindel and showed up to help?  Could one of them have received one of the Three Rings?  Or more, could one of them have been the Ringbearer?  Aragorn would almost be unnecessary as a member of the Fellowship; Beren could have done the job equally well if not better.  (Or he could replace Legolas, or whoever else.)  I don't know, maybe they're ruled out for whatever the reason Glorfindel is ruled out.  This really takes us into Alternate Universe realms, and is hard to answer.  But I think it's safe to say that a lot of things would have been really different if Beren had stayed an Elf.