Where does the idea of reincarnation come from? Are there any verses in the Bible in favor of this belief?
With the increasing popularity of eastern religions in the west, reincarnation is becoming a more commonly held belief among Americans, Britons, and Germans--no longer only among Indians.
Actually, Hinduism has taught reincarnation for many thousands of years:
Accordingly, those who are of pleasant conduct here -- the prospect is, indeed, that they will enter a pleasant womb, either the womb of a Brahman, or the womb of a Kshatriya, or the womb of a Vaisya. But those who are of a stinking conduct here -- the prospect is, indeed, that they will enter a stinking womb of a dog, or the womb of a swine, or the womb of an outcast. -- Chandogya Upanishad, 5.10.8
This Hindu passage teaches that the body in which you live your next life will depend on your conduct in the present life. Karma (your deeds) determine your fate from life to life. This is the doctrine of works if ever there was one! And yet the scriptures deny this doctrine (Hebrews 9:27, Job 7:8-10).
Sometimes it is claimed that John the Baptist was Elijah reincarnated. And yet the scriptures indicate John came in the same spirit and attitude as Elijah (Malachi 3-4; Matthew 11:14; see also 1 Kings 17-19) -- yet he was not Elijah (John 1:21), not literally.
Moreover, reincarnation involves the rebirth of a dead person. But, technically speaking, Elijah never died (2 Kings 2:1-11). On top of this, Elijah appeared along with Moses at the Transfiguration (Matthew 17, Mark 9, Luke 9), so he was never "reincarnated" as John the Baptist!
Thus we see that there is no biblical basis for importing the popular eastern idea of reincarnation into Christianity.