While there is a rare Hebrew word "Shai" that means gift, this name does not derive from that word but rather is a short form of Yeshayahu, which is the name Isaiah. Shaye has been a nickname for Isaiah among Jews for centuries. [noted -ed]
Shai is also the concept of Fate in Egyptian mythology. Therefore you could use this name as a replacement for Karma or Destiny. Because the Egyptians saw Fate as being without specific gender (sometimes they called it male, sometimes female), you could use this as a genuinely unisex name.