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)
Other mistake: When Robin Williams starts to download Banister's life files the computer says there are 544,628 hours of footage (approx 62 years) yet shortly after a newspaper headline states that Banister died aged 54.
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.