If "habari" means "news," then does "Habari za asubuhi?" mean "What's the news of the morning?"
I'm probably wrong, but from what I took you can use Habari to ask question greetings. Like: Habari ya leo? ( How are you today?), so I would guess Habari nzuri to be kind of a loosely translation of (How are you?, How have you been?, Are you good?) Hopefully someone else can give you a more accurate or 100% positive meaning with this phrase.
No, habari nzuri is literally translated as good news (habari = news, nzuri = good). The reason why it's used in greetings is because when you ask "habari ya leo?" you're actually asking about today's news, but it is used as a greeting rather than literally.
Habari? (News?) Habari ni Nzuri. (The news is good/well.) So if I recall correctly, it's used as a way of asking how someone is doing, but the news bit would be the more direct translation. So Habari nzuri would be "good news". :) And something like Habari ya leo would be "what's the news of the day?" Hopefully I'm recalling correctly, haha.