Thread-Cutting Labrys
This labrys, or double-bladed axe has the power to 'cut the threads of Fates' that reverse divine provisions (Tithonus's immortality) plays a major part in Asterion's past. He guarded it in the labyrinth, used it himself on Tithonus, and met his demise under its blade too. I believe that its the remnant axe head you see displayed in the hotel's depths.
Speaking of blood:
Only those with divine blessing or blood are able to wield it without hurting themselves. Laomedon got his hand burned by attempting to do so, while Asterion and Athena's hero was able to make use of it. This is why Asterion cautions us against touching it and was fairly confident that Clement was not able to tamper with it (7th, 8th, 17th tablet)(2nd visit to the hotel's depths).
A liquid drips from the artifact. Probably ichor. I think the ichor belongs to Tithonus (he bleeds ichor due to Zeus's blessing) in particular. His death was a sacrilegious act, and it was remarked that the ichor won't wash away. In fact, Asterion remarks that the unwashable mess profaned the shrine where Tithonus was executed. Meanwhile, Asterion's beheading was deemed by Olympus to be a correct act. This 'unwashable ichor' brings up some interesting possibilities. Asterion said that it rendered a god's shrine "no longer a place for worship or a sacrifice". Could this be something that can be potentially used to nullify or get rid of divine properties in objects? Think of the implications!