Seems likely that 343 left her name unknown to see what fans want before making a choice. But I don’t see much talk about it since people are busy discussing other things.
So give a vote and maybe give some suggestions for what names could fit her.
0 voters
I’ll share some of my own thoughts.
I wouldn’t mind her naming herself Cortana. It’s nostalgic, somewhat makes sense in the story, and allows me to kinda maybe sort of pretend Halo 5 never happened cough.
I have also gathered some cool info that may or may not be related to her name, but nevertheless is really interesting.
Firstly: Bungie’s two AI characters Durendal (Marathon) and Cortana (Halo) are actually connected.
Their names are based on a poem called “The Song of Roland”. In this poem there are 3 swords made out of the same steel: Durendal, Cortana and Joyeuse.
If 343 decided to do a callback to this the weapon’s name would be Joyeuse.
In my opinion not the most fitting name but I thought I’d share that piece of Bungie AI lore at least.
Secondly: I heard from Harusbis on the Sacred Icon podcast (you may know him from one of his blogs, lore-sites, or the waypoint canon-fodder).
He said that there has been some speculations regarding a connection between Cortana and Vinnevra (from the book Primordium, 2nd book in the Forerunner trilogy). Based on some interesting similarities fans have picked up on.
In this book we learn that Vinnevra’s “real name” is actually secret, and we never learn what her real name actually is. Some fans have theorised that her real name could have been Cortana.
So what if the weapon named herself Vinnevra?
Just some thoughts, take it with a grain of salt.
Give all your suggestions, thoughts and opinions!
For people who the Cortana name doesn’t sit well I offer an idea that will make it a bit better:
She can be named Cortana without actually being Cortana, a lot of kids end up with their parents name, and it kinda feels like shes a more innocent version of Cortana.
My bet is on Contessa unless thats a name for an AI I’m unaware of
There are no current AI’s with that name (at least in Halo). What makes you drawn to that one?
Its part of the name of the Orange/Peach coating from the holiday event called Sunset Contessa. The description says “An ONI codename now known only to Cortana.”
https://www.haloinfinitenews.com/armor-customizations/82c8c8ca-3916-45a9-8707-2395a0df42f4
Ohh, that’s very mysterious. Interesting
I’m just happy I share my vote with the majority.
Named after an AI who went full on genocidal and killed billions? Na.
Let’s assume there’s no funny twist business down the line.
The Weapon is a Cortana model AI, so that is the name Cortana chose so it is the name she would choose.
But you asked me should. Well it’s kind of wrong to imply that you could just replace her with another character. The game kind of suggests that because the Weapon is so like Cortana this is a second chance for Chief to get things right. But it’s not. In Halo 4 it’s a case of saving your friend who you’ve been on adventures with for ages; not a case of making sure the Chief has an off the shelf AI companion. You get given the Weapon at the start of the game so it’s basically a static character and there’s no real room for drama. She exists entirely in the shadow of this other character. Without those memories and that connection all that won’t transfer over automatically. She’s an entirely new character.
Plus it’s a bit like Wanda trying to make another Vision in Westview. It’s not right for you or the Chief to pretend this is Cortana. Normally if you have a story about grief and moving on; it’s not with a clone. That’s not moving on or making things right.
Also, why does the Weapon want to be Cortana? It’s kind of weird to want to be another guys dead partner. Yes, sci fi Logic and she’s not a complete blank slate. Perhaps it’s more to do with her accepting that’s what she is. But I would have said a blank slate can choose almost any identity really; much less name. She’s the one choosing to talk and look like Cortana. Quite an easy fix for a hologram. 
Because she’s not Cortana it would come across as weird if you start calling her that. Either it will feel like part of the committee retcon or it will be a constant reminder that it’s not really her.
Logic Plague.
She also saved the galaxy a few times and spent half her life making sure the Chief stayed alive in the Forward Unto Dawn. The Weapon taking Cortanas name is her trying to emulate the Good in the character and own it. If you’re going to be called that anyway and defined by it; you might as well start setting the terms.
The Arbiter destroyed Reach and killed billions of innocent humans. We forgave him. We forgave the Elites once the stopped their massacres and accepted peace. Something the Banished and Brutes have not done and seem mentally incapable of understanding.
Yeah I do not understand 343 insisting on throwing her under the bus (again) with Doisac to make a faction of cannibal alien fascists sympathetic. The Banished and Atriox are horrible. They don’t even use Doisac as an excuse for their behaviour. Eshcarum even says it’s a good thing because it made them stronger. These nutcases challenged a God they had no way of beating and got stomped. Atriox is not smart. Can you imagine if they had killed the Chief and failed to kill Cortana? What do you think would happen to the Banished and Brutes then? Since we’re talking about unintended consequences here Id say that’s a dumb risk to take.
I vote her name to be Cortina or Cor2na or Anitroc
What about CoR2D2na? 
It would be to weird to name her Cortana, the real Cortana died in Halo 4, the rogue Cortana didn’t deserved to be call Cortana and that only curse her name and legacy. Now everybody who ear the name of Cortana only think to all atrocity she made.
The weapon isn’t a clone of Cortana, she’s more like a sister, born of the same mother. Have you ever see two sister, even twin, with the same name ?
I felt like this was heavily implied at the end, when she asked Chief if it would be okay, it was damn near the most emotional moment in the game (alongside Spartan Griffin’s death.)
Name her Infinite. 
Joyeuse. Joy for short.
Fair point. But the Arbiter was an enemy who became an ally, not an ally who betrayed his people. It’s debatable as to whether or not everyone forgives the Elites for their genocidal campaign. Even Lord Hood stated he couldn’t forgive them at the end of Halo 3. Doing good things in the past doesn’t absolve genocide in the future. And to be honest I feel slightly vindicated. I was in the minority in the community because I supported Cpt. Del Rio when Halo 4 first came out. 
Cortana is her own singular character with a unique personality. If your trying to establish a new character with a different personality it would be best to have a different name. Not saying two characters with the same name can’t be well written but if it’s simply for the sake of nostalgia that’s just lazy writing. I mean it would be better if we didn’t have to refer to these characters as Cortana 1 and Cortana 2. It seems to diminish the significance of each individual characters uniqueness. I’m definitely a Weapon fan and would love to see her grow into a significant character based on her own achievements and merits and not to have those conflated with Cortana’s. She should be her own person or A.I.
If Del Rio had destroyed Cortana the Didact would have composed Earth and all humans in the Galaxy.
If there’s time travel and Atriox big brain plan is to go back in time to kill Cortana. That means either the Covenant or the Flood wins.
Did the Geth betray the Quarians in Mass Effect? Humans at the end of the day keep AI as slaves, which they can kill on a whim and this allows them to benefit from their existence. The UNSC is just as much a slave economy as the Quarians had and should have the exact same issues. Especially when mandatory killing of any AI is standard practice. It’s only because of the 7 year lifespan, exactly like Bladerunner, that they’re able to avoid the issues of an AI Singularity.
Well I am still not sure why the UNSC didn’t blockade and bombard Doisac with nuclear weapons. If the Banished were launching genocidal raids on human colonies for decades it’s odd they didn’t take retaliatory measures on the Brutes. I don’t recall the Covenant offering unconditional surrender as an option.
The UNSC retains a full nuclear arsenal. If it’s like the real world then it’s fully prepared to use these on civilians targets. This includes planet cracking nukes like the Nova Bomb in the books. Never mind the Covenant destroying planets. So humanity can’t claim the moral high ground and definitely not Space Genghis Khan. Chief really shouldn’t be shocked since he’s essentially a member of a Government Death Squad and they could have asked him to plant nukes in rebel human cities.
Also, if the UNSC has destroyed the Domain and killed all the Created. That’s kind of also genocide. Most sentient AI joined Cortana. That’s no different than the Quarians killing the Geth and just for the record the Geth killed far more Quarians than humans have been killed by the Created.