The scenario of the New England states breaking away from the USA in the early 19th century has been brought up many times and timelines usually give it a name like Republic of New England or something similar. It's always struck me though that so soon after the Revolution it seems very unlikely that this new nation would choose a name with "England" in it. As a region its a fine name but different considerations come into play when naming a country.
On the contrary, it was already a well-defined region early on during the colonial era and was in fact widely embraced by the Puritans themselves. Something like that cannot be erased overnight. So I agree with
@The Tai-Pan - as it was already established as such as a regional name (and the few times it was considered distinct, like the New England Confederation military alliance and the failure that was the Dominion of New England), New England can easily remain as the name of a country. Heck, one can go further than this and take New England literally as a new and improved version of what Old England could have been (and thus harkening back to the Cromwellian period, which so many supported), thereby already mentally decoupling the second word from the area it rebelled against and going for more positive political connotations. Furthermore, as Massachusetts was the most dominant state and its leaders often took on the term "New England" for itself (although causing much consternation elsewhere - see the analogous terminology confusion in the UK itself over "the UK/Britain" vs. "England"), New England is probably more neutral than just simply appropriating "Massachusetts" or any other potential options which would be seen by others as favoritism for one side or state over another.
In other words, TL;DR - if it ain't broke, don't fix it. That simple explanation is why TL authors almost always consistently use "New England" (with varying formal official names) as the name of an independent country. There's no reason to change, though not for lack of trying but because as far as people in that period knew there were no other alternative options. As a result, for my contribution to this question:
So assuming New England breaks away from the USA sometime pre-1840, what are the potential names for this new country?
I'd go for a variation on a theme, since there's other names for "England" that could be used without specifically tying it to Old England. New Albion, for example, or New Anglia (playing off of the historical roots of the colonists in the region of East Anglia), or if one wanted to go down a more Arthurian route (not that anyone in Puritan New England would be familiar with the King Arthur myth during the period in question) New Loegria or New Avalon.