Just my hapence, Hamsa means five in Arabic. I don’t think you can accuse Jews of making it up to sound less jain. Ahimsa means compassion or something in Hindi, I think — not sure if it’s somehow related to the word Hamsa.
OhTeeDee, I see what you’re saying. The Zohar was according to tradition written by Shimon Bar Yohai in Talmudic times and published for the first time in Spain in the 1200s. I’m not familiar with the cultural context of the time, but I’m fairly sure Hinduism and Buddhism were not popular there at the time — my guess would be that they would be completely unknown. Hinduism was traditionally confined to India, and Buddhism never became popular further West than Afghanistan. The Ramban’s mention of gilgulim had to do with his explanation of yibum (marrying your brother’s widow), and I don’t see any connection of this to Eastern religions.
It’s true there’s no obvious reference to gilgulim in the Torah, Tanakh or even the Mishnah or Talmud. However, it is conceivable that belief in reincarnation was passed down from teacher to student in small kabbalistic circles, and only revealed in the 1200s when someone (Hashem?) decided that it was time to reveal these teachings to the world. It is also conceivable that gilgulim are real, but that they weren’t part of the original belief system of Jews at all, until by ruach hakodesh and other spiritual experiences great tzaddikim like Shimon Bar Yochai or later figures realized that gilgulim is part of how Hashem runs the world.
Keep in mind that many things that are part of normative Judaism today are not found in the Torah or Tanakh (detailed descriptions of gehenna, olam haba, lighting shabbat candles). We rely on not only the Oral Torah, but also the teachings of the gedolim and tzaddikim in all the subsequent generations. If gilgulim is heretical or not true or whatever, then why did virtually all rabbis come to accept it within a few hundred years of the publishing of the Zohar? Just a thought!