Funerals are a great cause for contemplation. I believe that your friend's or loved one's life is honored by the question you've asked here. When it comes to matters of death and immortality... we don't have much to go on but faith. Because, there is no scientific evidence (that I'm aware of) that proves with reasonable certainty that we even have a soul. But I like to believe that we do have a soul.
Most Christians that I know (and I know a lot of them) interpret the bible as saying that the soul is immortal for both saints and sinners, believers and non-believers. Not sure if that is a matter of fact... or just a better way of scaring the hell out of people.
But the farther outside the bible you look, you'll probably find more and more disagreement on this question.
For instance, you're idea that believers have an immortal soul and non-believers have a mortal soul and a chance for a do-over... reminds me of Hinduism. I think there are several major religions that paint a less "terminal" and punitive outcome than that normally preached by Christians.