when does it become "named game"?

If you have ever used the Game Explorer (or any database), you know your first choice say 1 e4 may show 698,000 games and then each choice thereafter (let's say 1.. c5) that cuts the number quickly to 318,000 and then 2 Nf3 has only 255,000 but then 2..Qa5 cuts it to only 32 games and a few more moves from the beginning you reach the status where you have very few individual games, perhaps Joe Vs. schmo 1851 and the next move takes you into the territory of the novelty.

Some lines take 25 moves or more to reach the end of the known line. Others only a few.

Because the moves are so bad, it's very easy to play a chess game that has never been played. For example. 1: a3 h5 almost certainly has never been played in a game and it's not even move 2.

But maybe I'm wrong.

If your talking about an official tournament game, there doesn't need to be any moves made at all, and the game will still be recorded in tournament history. For example, sometimes a player doesn't show up for the game, like Bobby Fischer in one of his world championship games against Boris Spassky. Even though there were no moves made, the game is still in recorded history as a loss for Fischer and a win for Spassky. It's also possible for the first player, white, to make a move, and black might resign before even making their first move. And the game would still be recorded as an official tournament result, and game score. I think this is what the op was asking.