Another theory, maybe in favor of Snape/Lily
Quoting "Anelli, Melissa and Emerson Spartz. "The Leaky Cauldron and MuggleNet interview Joanne Kathleen Rowling: Part One," The Leaky Cauldron, 16 July 2005":
ES: This is one of my burning questions since the third book - why did Voldemort offer Lily so many chances to live? Would he actually have let her live?
JKR: Mmhm.
This is a very interesting answer, I think. So Voldemort really tried to save Lily when he tried to kill poor baby Harry. Why? He didn't bother with James though.
Could this be another point for the Snape/Lily theory? Maybe Snape made a deal with Voldemort to let Lily live, or even just to try to, in exchange for something else? If Lily had survived, she would have had a life debt to Voldemort, or, in the case this theory is true, to Snape.
I don't think Voldemort planned this whole thing (like: ""I'm trying to establish a life debt between Lily and me") because we are repeatedly told that he doesn't care for "this kind of magic", the emotional bonds between people.
What if Snape was behind it? Any further ideas?