The Coelacanth is the rarest fish you can catch at the docks. The short version: Fish with a Super Lure using a character who has high Luck.
Some ways to get luck: Winter has the highest luck, then Ryan, then Madilyn. There's a wishing well in the forest that can give you a cape, a cap, and weapons that all boost your luck. You can also equip rings that raise all stats, or the charm rings bought from Beshevaak at the desert pavilion.
Looking back, I made this one pretty hard to find. Fishing has a 30% chance to start a fight, then there's another luck roll to see what tier of item you get, then a final roll to see which item. So even if you have enough luck (250 minimum, 550 to guarantee best rewards), there's still a chance you get a fight or a rare weapon instead of the Coelacanth. Good luck, and thanks for playing my game!