I would've said the same until I read the 'Green Jay' comment. I would still prefer no codename over green jay but if she absolutely must have a codename, that might be okay. Her phoenix costume was mostly green so maybe going back to it can still work. She could go with the bird theme still and since she is no longer a phoenix, jay would be as good as any.
Though as many lointed out, probably won't stick.
Phoenix kinda doesn't make sense even though she has used it before without the PF since she outright rejected it recently.
Marvel Girl or Woman would be too close to Captain/ Ms. Marvel and if they are pushing these 2 to be their top billed female characters, would feel too bland to the non comic reading audience.
Overall, Jean Grey is fine.