Both "the day is young" and "the night is young" go back more than a century as idioms or truisms. Consider this excerpt from "," in Blackwood's Magazine (August 1824):

Good heaven! somebody will say, what is the meaning of this rigmarole cock-and-bull sort of nonsense? Do you take us for Peripatetics? By no means, my good friends, but there is no need for hurry. The day is young. Hooly and fairly goes far. Take the world easy. Blow not your horse in the morning, and you will be the farther on when night falls.

And from Arthur Reed, "," in The Cambridge Review (May 23, 1889):

Our griefs shall break like a bursting bubble, / Our fears like a curl of smoke be gone; / So drain a bumper to sink all trouble, / And pass the punch as the night wears on! / Then drink all round / Till our heads be drowned— / The night is young and the bowl is crowned!

Here is an Ngram chart of the "the day is young" (blue line) and "the night is young" (red line) for the period 1800–2008:

The chart indicates that the "day" version came into currency before the "night" version did, but it also shows that, since about 1950, the "night" version has been considerably more common that the "day" version. Still, both expressions continue to appear fairly frequently in published works.

Of course, when people say "The night is young," they generally mean "I'm still having fun and I don't want to go to sleep yet"; whereas when people say "The day is young," they may mean "There is lots of daylight left and lots of time to do what we need to do" or "We've awakened near dawn and the whole day lies ahead of us" or even "Okay, it's after midnight, and it may be inaccurate to say that the night is young—but if it's only 1 or 2 or 3 a.m., we can certainly say that the day is young."