In my opinion, the frostjaw had the potential to be a lunar counterpart to the Daywalker.
Locating - The Daywalker is always located in the muddy biome, which is a sort of central-ish location, and is thus connected to the ruins most times. The lunar frostjaw should be situated near the lunar island, both as a way of making it easier to locate, and also making sense in the lore due to it needing to be close to the lunar island for the lunar power to take effect.
Spawning method - In the case of the daywalker, you need to go to the ruins to get the pick/axe to shake the pillars to fight it. To access the lunar frostjaw's fight, you should need to go to lunar island, craft a bath bomb, and place it in the middle of the frostjaw's ice platform, where there would be a crack of some sort. The bomb would then fizz away, leaving a hole in the ice for you to fish out of and start the fight.
Design - I envision that the lunar frostjaw would be mostly the same as the regular frostjaw, but instead with some enlightened ice. I have made a crude mockup of what it would look like out of enlightened shards. (keep in mind this just a concept, sorry if it doesnt look that good)
The fight - Nothing really to say about this, maybe buff the health and damage a bit to put it on par with the daywalker, and also because of the new drops.
Drops - Once the frostjaw is defeated, the lunar ice will break off, as well as some pure brilliance . This ice would also be able to be periodically mined from icebergs surrounding the island. This new lunar ice could be used, along with the pure brilliance, to make a new armor set called frostbite armor, which would essentially be a lunar counterpart to the dreadstone armour. In addition to this, the ice could also be turned into a rain beacon that goes up into the sky, and makes it so that nearby rain will all be frozen on it, giving a solution to rain, in turn for making it colder in the area. The reason that this would exist is due to it being a counterpart to the dreadstone pillar, which blocks earthquakes, and earthquakes are essentially just cave rain.
Drops (extended) - The frostbite helm would look similar to the eye mask with all of its teeth, and the frostbite armor would be a mix of the void cowl and dreadstone armor, with the lunar colout palette. as for how it would work, I feel like it would restore durability over time while over 70% sanity, but it would also put you under the effect of enlightenment while wearing it. This means that to repair it over time, you would need to deal with gestalts.
Overall, i think there was a lot of potential with the frostjaw being a lunar counterpart to the Daywalker, and that potential was wasted.