I keep seeing people think she’s gonna name herself Cortana. There’s no way.
She saw first hand throughout the story how much Cortana means to Chief, and how hurt he still is from losing her.
We don’t even get to see Chiefs reaction to Cortana’s message/goodbye at the end of Infinite. And Weapon doesn’t even know what to say to him to ask if he’s okay, cause she knows he’s not. Last thing she’d do is put sand in the wound and make him address her as such.
Not calling herself Cortana ≠ Not replacing Cortana.
But I think she’s either gonna call herself Miranda, or name herself after another sword (curtana is a sword I think).
I think she should be called Miranda
Little Miss Perfect. 
How does it NOT lead to ANYTHING BUT naming herself Cortana?
It could literally not possibly be anymore obvious.
The association with a psycho tyrant that recently commited attrocities?
But she asks the Chief if it’s okay. Yes, she could just be asking if she can name herself; but it’s very much framed as a “can I take her name Chief”.
It’s meant to be her honouring the good in the character which she just saw and partly coming to terms with who she ultimately is.
Possibly the writers were leaving it open to gage peoples reactions to the character. I mean, aside from a few mild memes I haven’t seen much of anything bad said about the Weapon.
It would get confusing if you have two Cortanas. Especially since they are probably going to continue referring to the previous one.
Chief gave her permission to name herself, not name herself Cortana. And like I said, Weapon isn’t as crass to do something like to him after seeing what he’s going through.
Nor do I think Cortana is done in the story with how abrupt/off screen her ending was Infinite. Her and Weapon will be two separate entities as characters. Naming them both Cortana just makes everything confusing.
Another reason that she may not name herself Cortana is because she isn’t. Just like how the AI’s are not the people the brains are scanned of, I do not think the resulting AI (even if from the same brain) would be the same or want the same name as another AI. The weapon is more like a sister to Cortana than anything else. She will probably give herself another name. kinda weird this wasn’t settled in the game.
But she’s essentially a clone of Cortana. Her without those memories. It’s not exactly a normal situation. If that’s the name Cortana decided to call herself then, a blank slate version would likely pick the same name.
I don’t know. A second fake out with Cortanas death could be a risk. If they really wanted the character alive in some way they could have said “oh she merged with the Weapon” or some fragment of her survived. They really seem to want the Chief to have taken this loss because of Atriox.
Given that so much of the game focused on Chiefs failure and the Weapon it’s as presented. The Weapon is Cortana 2.0.
Let’s assume no time travel for now.
Even if she’s a clone, she’s not Cortana though. It’s the same way Cortana isn’t Halsey even though they share the same genetic makeup, she has her own personality that’s been created because of what her and Chief went through together.
What makes the most sense is for Weapon to name herself considering herself as like a Sister, not Cortana herself.
And Aatriox isn’t taking his loss from Cortana though, why would Chief. The only reason he would be allied with the Endless would be to save his people. And what Chief wants, (saving Cortana before the she’s killed by the Didact) literally goes hand in hand with what Aatriox wants.
In that manner her death isn’t really a fake out. H5/Infinite Cortana still gave her life and isn’t coming back.
I don’t consider it a coincidence they highlighted Cortana wiping out Aatriox homeworld, as well as all the Spartans she killed if it wasn’t something that was going to be addressed/rectified later. Cause that’s some heinous stuff for them to just off screen then end her character on.
Oh don’t get me wrong, I agree, she is not Cortana. Like, I am not lying, whenever she speaks you’re immediately reminded Cortanas dead. When Cortana says in Halo 4 they’ll just get you a new model the whole point of that is that the bond they share is unique and you could never replace Cortana. However, in Infinite they explicitly go against this sentiment. The Ghost of Cortana says this new AI has so much potential and can learn from her failures. That they would automatically be perfectly perfect.
Now, I could counter that and say that without that bond the Weapon would be more, not less likely to fail and fall to the Logic Plague. But I don’t think that’s the intent. It’s that Chief failed and his mission is to make sure she doesn’t.
I reckon they are going to make it personal between the Chief and Atriox. They make a point of him being beaten, that being why doesn’t get to Cortana and why she had to sacrifice herself. She is very much dead because of Atriox. Part of that would involve the character being dead.
As an aside there’s no indications Atriox is a good guy. He butchers the people who actually stopped Cortana and prevented the near certain genocide of his species. His actions show he does not actually care about Doisac or the Brutes and is just a hypocrite out for power. He’s a tyrant leading an army of cannibalistic aliens who want to burn Earth out of twisted revenge.
They did the evil stuff to set up that the Weapon is afraid of becoming like that. Which is being established as a danger which the Chief needs to be aware of and figure out a way to avoid. I mean, as far as we know the Domain is still there so it is something that could happen. No idea how far 343 wants to push that and if the end goal is she becomes a good and benevolent version of OP Cortana. But yeah, it’s done to establish that danger and tension between the characters.
I wouldn’t be opposed to them bringing OG Cortana back. I just don’t think they will or intend to given what they’ve done.