That’s a half-myth it seems. There was a write in campaign from the fans to rename the then tentatively named Constitution (in recognition of the 200th anniversary of the ratification of the US Declaration of Independence), to Enterprise. At the same time, Gerald Ford had the final say, and he himself served aboard the famous WW2 aircraft carrier, the USS Enterprise (CV-6). It would appear that there is a confluence of two wishes to lead to the test vehicle being finally officially named the Enterprise. Certainly of recent years, NASA has seemed happy to not dispel that story though.

Some interesting tid-bits here though. The NCC-1701 was early on going to be named the USS Yorktown. Yorktown-class was the ship class of the CV-6 Enterprise. While we all know that the ship class of the NCC-1701 was the Constitution-class.

So, it’s my belief therefore that, in our timeline and Star Trek’s timeline, American men of a certain age seem to be obsessed with the symbolism of three particular names; and no matter what, the Shuttle would have been Enterprise in both.