The 8-bit Zelda games are about a character named Zelda who was put into a never-ending sleep because of her brother. Their father the King passed on and his children only inherited two pieces of the Triforce, the third was hidden away for a reason. A magician close to the King informed the Prince that the King told Zelda something about the missing piece before he died, so the Prince interogated his sister but she wouldn't speak. The magician threatened to put her into a never ending sleep if she didn't tell him what he wanted to know, but she wouldn't say a thing. The magician cast the spell, but the Prince didn't want that but couldn't stop him either. After the spell was cast Zelda fell on the spot, and the magician breathed his last. Stricken with grief, the Prince placed his sister in an alter until she awakens, and so that the tragedy is never forgotten he ordered that all females born into the Royal Household shall be named Zelda.
In Zelda II, Link embarked on a quest to find that missing piece of the Triforce which awakened that sleeping princess, the first Princess Zelda.
Hurrah for timeline theories!
This. To add on to his point, in the game of Zelda 2, when this story is revealed the characters call this story "The Legend of Zelda." So the name of the game is after the actual name to the actual legend of the first Zelda. Of coarse the story isnt told in the game and the name of the tale isnt technically stated in the game, it is however told and named in the games manual.