Q. Is there in Jewish tradition a reason or purpose for shaking the hand when saluting or greeting someone?
A. Handshaking, according to ancient texts and testimony carved in relief-slabs from ancient Greece, was a common greeting practice millennia back. According to some it was originated as a gesture of peace, indicating that no weapons were on hand. However, not much of it is to be found in Jewish tradition.
The greeting is commonly described as "nesinat shalom" or "giving" shalom, which could be interpreted as an actual giving of a handshake. ("Ragil liten lo shalom", Brachot 6b, "Hanoten shalom lechavero", ibid 14a, "Venatan li shalom", Avot 6-9, and many others). Then again, the same expression is used in regards to Hashem, as in "Klum yesh eved shenoten shalom leravo" (Shabbat 89,a), where a physical handshake is unfeasible.
Sefer Divre Torah (9,14), relates that Mahara"sh Alfandri ZT"L usually refused to salute with a handshake, as he considered it to be a new European trend, unknown and strange to them. Conversely, Megadim Hadashim (p.14) quotes different sources that indicate that it is not only correct to salute with a handshake but actually desirable. Bnei Isoschor (on Mishlei 11,20) mentions that "it is a tradition by us, that if someone arrives from a journey to give vehemently shalom with a strong handshake". Similarly Tiferes Shlomo (Moadim- Smini Atzeres) advices to greet by giving his hand, because "whoever blesses his friend and wishes him goodness, the blessings are in his hands, and they are transferred from his hand to his friend."