There's a redemption arc to the story. Extreme destruction and self sacrifice (for the right reasons). Wanting to yet be called Cortana, signifies that beings can change for the better, especially when given second chances.
It's also pointing out not to rush to judgement. Something both the Weapon and Chief have to struggle with and learn to overcome. There's a time and place when immediate decisions and action must take place, but those times are more rare.
If she's going to remain unique instead of showing at least some of the good or harmless traits of original Cortana, then I'd personally rather have a different name as well, but it would be totally understandable if 343 sticks with Cortana.
PS1 - ! - We must build a console that can alert our enemies.
PS2 - @- We must build a console that offers online living room gaming.
PS3 - #- We must build a console that’s powerful, social, costs and does everything.
PS4 - $- We must build a console that’s affordable, charges for services, and pumps out exclusives.
PRO -%-We must build a console that's VR ready, checkerboard upscales, and sells but a fraction of the money printer.
PS5 - ^ -We must build a console that’s a generational cross product, with RT lighting, and price hiking.
PRO -&- We must build a console that Super Res upscales and continues the cost increases.
