I read this story in OMNI when I was a kid sometime in the '80s. The generation ship has no living crew, but will create one person from DNA when it gets close to a candidate planet to determine if it's right for settling. If it is not good, the person has to live out his life and die on the way to the next planet. This has happened multiple times, and we see the latest version watch the logs of what his previous lives did during their turns.
Another couple details I remember. It's always the same guy and I'm fairly sure he has memories from before the ship launched. His wife is available to be cloned/created too, and on several previous occasions he revives her and they live their lives out together.
I don't recall the month/issue/cover of the magazine.