Revealing mistake: Alan is watching his memories at the end after he figures out Louis didn't die; as young Alan is running out of the abandoned place you see the metal where he originally wrote his name in the beginning, but it was blank, the name Alan wasn't there. (01:18:07)
Revealing mistake: Robin Williams is working on Bannister's rememory, scanning newspaper articles. Under the first headline, "Bannister Eyes Future," the copy reads, "Whoever wrote these headlines had as much fun as I did setting them in type and formatting them into newspaper articles. But then, this sort of headline appears in the paper regularly, so maybe we all just want to have fun. Fun: such a precious commodity these days."
Answer: The movie provides no answer as to why the device was named "ZOE." However, one should note the word "Zoe" is actually a greek word for "life." Since the ZOE implant records someone's life, this sounds like the most logical answer to why it was named ZOE. On a side note: The first device to display moving images was called the "Zoetrope", the similar name could have had further influence as to why this implant was named ZOE.
Kelsey H.