How long have we been calling it that?
It could qualify if it persisted in that name for longer than anything else has before or since. It only needs to be continuous, not still in existence today.
****New Covenant:
The repertory of the Ars nova, a musical style of the 14th century, is still referred to as that, and the term was explicitly coined by contemporary theorists.
Nouvelle cuisine, the “culinary revolution” from the 1970s, coined then by two journalists, and which you still see thrown around, has been used by writers to describe their cooking since at least the 17th century.
I see your point, the expression was probably coined in the late 19th or in the 20th century.
According to the New 
World Encyclopedia:
Cenozoic (as Kainozoic) was first used in 1840.
The New Kingdom wasn’t called that at the time it existed. The terms Old, Middle, and New Kingdoms came into use in the nineteenth century.
If we consider things named after the fact, then Neolithic would be a contender, having existed from 10,000 BC to about 4,500 BC or later.
Well…if you’d clicked the link you’d have seen the 889 reference to Niunstadt.
Admittedly, it’s in dispute, but it’s a very old “new” thing.
In that case, yeah, disqualified.
Just to be really clear, I’m looking for things that lasted a long time along with their coined name containing the term ‘new’ - not long-lasting things that were later called ‘new’
Not quite a match for the New Forest, but notably old: New College, Oxford dates from 1379.
Ron Wood, the “new” Rolling Stone" is 68. Just saying…
I stand corrected :o (I was irritated by the fact that the coat of arms of the city was created in 1968).
Not really meeting the OP’s criteria, but perhaps of interest anyway: the Old New Synagogue in Prague, built in the 13th century. It was originally called ‘New Synagogue’, but they eventually got around to building newer ones. Nowadays, it’s actually the oldest active synagogue in Europe. Also, it’s said to contain the remains of the Golem.
The Emperor Constantine founded New Rome in 330 A.D. and, while the city is now called Istanbul, it’s still referred to as New Rome on occasion by the Eastern Orthodox Church.
But that might be a bit a bit of a stretch for the OP.
It’s not at all in the running, but I thought it deserved a mention.
Massachusetts’ New State House, which is still called that (and not simply " the State House") has been called that since it was built in 1798. The reason is that the Old State House (which is only 85 years older) is still only a couple of blocks away. The Old State House is still called that.
By comparison, Boston City Hall (only a couple of blocks from both of these) is no longer called “New City Hall” , as it was when I first came to Boston, and Old City Hall is rarely called that anymore. Now they call it Ruth’s Chris steak House.
Wikipedia says that Novaya Zemlya (“New Land”) was known to the Russians from the 11th century, but I don’t know how far back the name goes.
Carthage, meaning “New City”, is quite ancient. Some founding legends date it to just after the Trojan War (lots of events are linked to that in classical “history”). More reliably, it was probably around by at least 800BC.
Cartagena, in Spain, was refounded as another Carthage in the 3rd century BC, renamed Carthago Nova: New Carthage (New New City) by the Romans.
Cartagena, in Columbia, was named after the Spanish one. Some people called it unofficially New Cartagena (New New New City) to distinguish it from the old one.
By that same method of thinking, the New World should be dated to the Americas splitting off from Europe and Africa a few million years ago, rather than to the coining of the phrase, “The New World”.
Similarly, I could just call our universe “The New Universe” and now I’ve won the thread.