Enchantments have a lot of uses, even for your humble Fishing Rod. Lets go over them, and help you choose what may be right for your next arcane adventure.
Luck of the Sea
When it comes to Fishing Rod enchantments, this is likely one of the first to come to mind. Luck of the Sea has a maximum of three levels, each of which increases your chances of catching treasure items. This may be a useful enchantment if you're looking to snag a few extra valuables while fishing.
Lure
Lure is a simple enchantment with a maximum of 3 levels, each of which decrease the amount of time between bites on your line. A useful enchantment if you're either impatient, or if you simply want to gather as much food in a short amount of time.
Unbreaking
As with all durability-related items, Unbreaking assists with tool longevity. With a max of 3 levels, applying this to your Fishing Rod will substantially increase its lifespan. Recommended if using against hostile mobs or players.
Mending
Granting durability upon collecting XP, it's a great enchantment if you want a truly unbreakable tool. It is rare, but very much worth adding to any important tool, especially if your Fishing Rod has a max level Lure or Luck of the Sea enchantment on it.