[SPOILER WARNING] After beating the campaign

The AI and John are talking about her name. At least for me the discussion was about if it is okay if she choose the name Cortana. Why is she still named Weapon when you walk around?

Probably to reveal the name in the future.
I personally think it would be inappropriate to assume a name feared by everyone across galaxies just for the sake of nostalgia, and especially after coming to terms that you will never be Cortana.

If we have Joe Staten and if they are still a little of a Bungie head I predict that the Weapon’s name will be Joyeuse

Durendal from Marathon. Cortana from the Halo Series. They were both swords made of the same materials.
Joyeuse is a third sword also said to be made from the same material as the other two. It would be actually fitting in a way.

No, but we glossed over that, remember? Lol. She’s the exact same Cortana just without all the bad. Which is a narrative way of saying “Sorry for doing that in the last game”

That’s not how it looked like to me tho, especially with Cortana’s last scene saying that she sees a future for you (john) and her.

She already saw a possibility of a future with John and another Cortana model and was coming to terms with it and that last Cortana hologram was kind of like a “well wishes” thing.

It would cheapen all of the hologram Cortana’s messages to just go “lol I’m Cortana 2”
After all, the Weapon’s story journey was all about learning the secrets of the ring and learning about herself and her origins, not to basically be Cortana 2.

I’m talking about this one:

https://youtu.be/f0SSAI8TCcs?t=12526

She’s asking John if it’s okay. Sounds for me she’s asking him because he and cortana had a relationship and if it’s okay if she has the same name because she’s a copy of her (or maybe not. Maybe both of them more like siblings)