Rainstorm Duel! Poison Frog of Wennath
Unlike all of the other quests in this section, this quest is actually a Hunt rather than a side quest. I've decided to write a guide for it due to the difficulty of trying to get it to rain in time to complete the Hunt. To get the weather to reset in Final Fantasy 15 you'll need to either wait, camp or fast travel.
What I recommend for this quest is to travel to the camp site shown in my screenshot below and repeatedly rest there until you wake up one time and it's raining. Once it's raining all you will have to do is travel a tad bit to the north and slay the Gaiatoad required for the quest.
The Gaiatoad itself is pretty easy to kill and straight forward, once you beat it you won't have to repeat this hunt again. Sadly, you do need to beat this hunt at least once to collect the required stars.