The first crewed (1+ biological human) ship to leave Earth farther than the moon will have the word Enterprise or a derivative of it in the name.

Destination requirement: another planer or moon in the solar system solar body, star, extrasolar planets, space, or other target all count. And, it must reach at least a distance of farther than the moon's orbit with a living crew member. Excursions with no particular destination but which reach the minimum distance also count.

The point is: will the show Star Trek end up having naming influence on ships.

Indefinite timeline.