So Bungie and 343 after them have always had a habit of hiding things in plain sight or at least making you think about things. To me, Weapon taking the name “Cortana” is too obvious and when you think about it, it doesn’t fit.
We find out in the name cutscene that Echo-216 is Fernando Esparza. He asks Weapon what he should call her.
Weapon asks Chief if it would be ok. Like most people, I figured she was asking Chief if he was ok with her picking Cortana as a name. However, Weapon’s too thoughtful and wouldn’t go with Cortana since Weapon knows what Cortana represents to Chief. It would stir up too much past every time he had to call her that. Remember, he tried to delete her because he thought she was going to turn into another Cortana. The story then takes you through showing that she isn’t Cortana, even though she’s a copy, she’s her own AI. Her own personality.
However, Chief’s next line is “You get to choose your name”… so she may have actually been asking if it’s ok for her to name herself and, with the name I think she picked, she’s asking if it’s ok because someone else is sort of using it at the time. Again, making it easy to think she could be asking if it’s ok to go with “Cortana”.
Now, think back to all the times you encounter Cortana in the game, or what’s left of her. Weapon keeps telling you they’re echoes of Cortana. Fast forward to where she finds out she’s a copy or in a way, an echo of Cortana, herself.
One of the definitions of an Echo is: “A close parallel or repetition of an idea, feeling, style, or event.” also “(of an object, movement, or event) be reminiscent of or have shared characteristics with.”
Therefore, out of respect for Chief and Cortana, and the fact that she is her own AI different from Cortana, even though she’s a copy, she’s an “Echo”.
Thus, the perfect name for her is… Echo.
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