You could use the fishing rod for Puzzles in dungeons (magical fishes turn into keys, you have to feed an monster to open the door for you or put one fish into each water hole of the room to make a chest appear) , for an extra income (being the only way to earn rupees would make the game frustrating though), and for an extra quest (NPC wants special kind of fishes from you which you obviously can't find everywhere and gives you good rewards like heart pieces, a special melodie required to enter a special dungeon or a bottle).
Maybe you can even add the possibility to turn switches underwater with the rod or something (just let some sign say "I connected my pet to this long thing down there, please don't chain it off. It likes to bite! - Zora Merchant"
) simply connecting a catched fish with a cutscene of a switched lever. Lotsa possibilities, just felt like giving some input since you asked^^