In its early stages of development, The Legend of Zelda: Oracle of Seasons was supposed to be a remake of Zelda 1. In the final version, it kept the same dungeon bosses and many gameplay elements.
If you already know the 2nd quest of Zelda 1 by heart and you are looking for more challenge, Faria is a game that has gameplay elements intermediate between the first two Legends of Zelda (it could be considered as "Legend of Zelda 1-and-a-half"). In particular, Faria features five dungeons in the style of Zelda 1, but on multiple floors!
Ultima I: The First Age of Darkness is the start of a different, but equally successful series. The first game features many gameplay elements that are reminiscent of this game. Although Ultima 1 is older by 5 years, the MS-DOS remake is the same age as Zelda 1.
Neutopia and the sequel Neutopia II are games released by Hudson Soft on the TurboGrafx-16 platform. They have similar gameplay with the first Zelda game, however they do have their own features, like magic wands and the compass.