Putting aside the 'brain in a computer' argument so prevalent these days, here are a few more ways in which it is scientifically plausible for reincarnation to exist.
1: Emergency Teleport
The moment before your body dies, your body is scanned and you are reconstituted in another location (or time). Any injuries you may have sustained leading to your demise will be healed using your DNA as a blueprint. If you were suicidal, those memories would have to be erased to keep you from attempting to kill yourself all over again.
See: Freejack, Gantz, The Girl Who Leapt Through Time, Down and Out in the Magic Kingdom
2: Ghost in the Machine
When you die, your "spirit" is intercepted, captured, and compressed, perhaps within a stasis field. You can then be recovered in order to transfer your awareness and memory to a new clone body.
See: Freejack, Eureka, Multiplicity, Serial Experiments Lain, Kaguya Hime, MPD Psycho, RahXephon, Captain America, Age of Fire
3: Symbiotic Relationship
When you die, your "symbiont" is intercepted, captured, and compressed, perhaps within a stasis field. You can then be recovered in order to transfer your awareness and memory to a new host body.
See: Star Trek DS9, Stargate SG-1, Babylon 5, Federation of the Hub
4: Freakish healing
When you die, your freakishly mutant healing ability heals your body and revives you.
See: X-Men, Torchwood, Doctor Who
5: It exists already, but nobody told you.
In this option, you don't remember your prior lives merely because you haven't lived them yet. You are The First. When you are reincarnated, the universe will roll the dice, and pick another being into which you will incarnate. You will remember everything, but you will never tell yourself that you do. Then again, memory fades with age, so you will probably only remember the last few incarnations anyway, if that. After all, you were just a baby, you could have dreamed the whole thing.
Why would the universe choose to incarnate The First partway into the existence of universe? How can the universe even be this far along if you haven't lived all of those other lives yet? I didn't say there wasn't a lot of timey-wimey handwaving involved. I just said it's scientifically plausible.