I'm sure whatever new starship the crew gets, if it gets one, it will be instantly recognizable. That said, I wonder if it will be named, Enterprise and be NCC-1701A (or any other letter). I mean why do it? It made quite a lot of sense in the other universe. By STIV, Kirk and his crew were legends, and the Enterprise was probably the most well-known ship in the fleet. It certainly saw its crew through the wildest adventures any Starfleet crew probably had. The Enterprise in the other universe (apparently, we don't know, but the teaser tends to make one believe) didn't even get Kirk through his first five year mission. By STB, there have to be other ships of the class of the Enterprise, and they are more than likely still the top of the line starships. So, maybe in a fourth movie (God willing, and all), Kirk and crew get one of those, and it will be the voyages of the starship, Constitution, NCC-1710, or whatever name and registry.
Presumably STB is half way through the Enterprise's 5 year mission, but it needs to be remembered that Kirk and Crew, while their story is not as established as their Prime timeline counterparts, have saved Earth once from Nero, and can be argued to have saved it from a war with the Klingons, where if Earth wasn't destroyed outright, could have been devastated. Plus, I am sure we will get some dialogue about how Kirk and crew had many an adventure, and distinguished themselves. SO, the point of all that is to establish Kirk is Starfleet's golden child. If Kirk wants his replacement ship to be named Enterprise, then by virtue of the fact of his past feats, he gets it.
It also must be remembered that the proper name of all the various Enterprises is in fact, Enterprise. We use the registry to differentiate between them, but since there is only one in service at a time, there is no need to call them Enterprise-A, or Enterprise II, etc. The next ship will likely be named just Enterprise. As far as what the registry will be, well, it could be a different number, but I suspect it will follow the naming conventions established in the Prime universe, so it will likely have the registry NCC-1701-A. So, unless they are just going to throw all that out the window, I think that is how it will unfold.
The question for me is what will it look like. Kind of why I started the thread, in the first place
I think there is a good case to be made that if the alternate-Constitution class is now the mainstay of the fleet, then the Enterprise-A may end up looking the same. That is assuming that a constitution class was just finished, and in order to replace the ship for Kirk, they simply rename it (sort of like they did with the Sao Palo when Sisko took command, and renamed it Defiant).
Then there is the tradition that each Enterprise (except Prime Enterprise-A, of course), is usually the newest class in Starfleet, and usually among the first of its class, as well. We could see a whole new class of ship. However, I think it will not be too different from the classic Jefferies design. I don't think we will see something that looks like the Kelvin, or Reliant, for example.
Then you have the rumored (all but confirmed) USS Franklin. What if this is a ship that the crew uses to get back home? Imagine if it survives the home trip, and is repairable. It is a new design, yet resembles the NX-01 Enterprise, so there is still a sense of familiarity. What if it is overhauled, refitted (maybe we will see an element of the NX-01 Drexler refit???), and rechristened the Enterprise-A? That could be interesting.