It has been proven by Tolkien himself that Eru did permit the souls of slain Elves to be reembodied once their hurts have been healed in the Halls of Mandos. It has also been said that some, such as Fëanor, would be given a sort of punishment, to remain in the Halls of Mandos.
While we know that Fëanor is kept in the Halls of Mandos, what of the rest of the House of Finwë? One of the most depressing elements of the First Age is the fact that all of the House of Finwë, save Galadriel, and Finarfin, were slain. The possibility of their reembodiment is uplifting, seeing as it's a wonderful thought that perhaps this doomed family is together again in Valinor.
The only members of the House whose actions could have brought about the same fate as Fëanor are Maedhros and Maglor, who stole the two silmarils from Eonwë and his host. We know Maedhros died, though all we learn of Maglor is that he roamed the beaches singing of the sorrows of the Elves.
All other members of the house of Finwë were valiant Elves, (not to say that Maedhros and Maglor weren't!) and despite their doom in the First Age, were deserving of the happiness that Valinor would bring.
One theory regarding the fate of Fëanor said that when Miriel died, she eventually joined Vairë the Weaver. Perhaps Eru or the Valar, after time, allowed Fëanor to join the mother who loved him so much? And perhaps Fëanor finally learned to control his fiery spirit.
Reference:
The Silmarillion