I love the concept of the Challenge Log; I do believe there is room for improvement, especially on the "Challenge" part of it.

Imagine each week, in addition to resetting your static challenges, it also introduces new challenges directly related to Boss encounters. These challenges rotate each week to a new random set of say, 5.

Examples
"Wetter as it dries" : Defeat Leviathan while he is a performing a slam on the boat.
Reward: +Grand Company Seals, +Gil.
Extreme: +Tomes, +Additional Gil, +1 drop (A randomized item from Leviathan's Loot Table + others, such as a stock of resources, potions, Tomes)

"Common Cold": Defeat "That second boss in Admapor City that I don't remember the name of" with the party having sequential stacks of the taunt debuff. (1, 2, 3, 4 for example)
Reward: +Gil, +Myth tomes.