(Only addressing NZ => Aotearoa country name change. Not efforts at promoting the language, customs and local place names)
If, by "the Crown" it refers to the Head of Commonwealth and assuming NZ has anything like the relationship we Canadians have with the "the Crown", then the answer is, no, it wouldn't. This would be a decision to take by the New Zealand electorate, acting through Parliament. British royalty has extremely limited power to affect sovereign states in the Commonwealth.
If the Queen is indeed the intended meaning, as the OP states and a comment does as well below, which I highly doubt, then this is likely more a PR move to gain international attention by appealing outside of NZ or appealing to an authority that is unlikely to quickly dismiss it. Which kinda worked, considered the link is from CNN.
If, by "the Crown", the usage is British-style legalese similar to "Crown land", "Crown companies", "Crown prosecutor", meaning "the government" (or in the case of "Crown land" "public land"), then, yes, that's possible. And Michael's answer does a much better job than mine detailing the procedure. But at this level, changing a country's internationally-recognized name is definitely something that would require a lot more buy-in from the electorate than just a petition. With 16% of Maoris that seems like a hard sell. But, as a PR exercise, why not?
British Columbia is also getting "nominated" for a name change. Same reasons given, really.
Should British Columbia change its name? As we reckon with history, some say it's time. The operative word is some, the rejection in a poll was quite significant. Not to mention practicalities: there are multiple tribes (30+?) in BC, many with their own language, and with tribal areas covering small fractions of the modern provincial territory, so determining which language's historical name to use would be an unholy mess.
On the other hand, we have changed, successfully, some "pretty big" place names, like Georgia Straight => Salish Sea, Queen Charlotte => Haida Gwaii. The reaction has been a lot more positive, as there is a lot less communal identity tied to these names, as well as considerably less administrative impact. Queen Charlotte? Don't think she ever had anything to do with the place.
p.s. If I were seriously trying to promote the language I'd shoot for renaming geographical features/areas myself, if I was aiming to restore customary names, rather than anything that would some up on existing deeds, contracts or personally-tied identification. Plus, a lot of native names are a lot more poetic-sounding than some long dead Brit's.
p.p.s Note also, that, in regards to this question, at least in Canada, "the Crown" has a specific extra meaning when it comes to First Nations relationships - "the Crown" is the formal counterparty to most of the original treaties natives <=> Canada and indigenous people here tend to be more attached to its participation (and somewhat distinct from "current Canadian govt") than the average Canadian might think. The fact that early treaties made in Canada were often more advantageous to natives than the latter. At least that's the claim made by Canada's Odyssey, by Russel.
Yes, I realize that Michael already said this, but I want to stress that the letter and intent of formal treaties originally signed by the Crown, are often appealed to when it comes to securing better outcomes for indigenous people, because later power relationships typically evolved to the detriment of the natives. So, no, not Head of Commonwealth, or even Head of State of NZ, but "respect your treaty obligations".