Empathiom (共感子, also translated as empathion and empathium) are a sort of energy source that the Travelers are seeking for their creators. The world of Theldesia is ripe with it after the Catastrophe, an event that the Travelers detected and thus purposely involved themselves with in order to harvest it.
In the world of Theldesia, Empathiom is linked to MP, the soul, memories, emotion, and the like. It takes the form of the shimmering rainbow fragments that appear from a person's body when they die.
While the true nature of empathiom is unknown, it is very likely that it takes the form of memories—particularly the memories sacrificed by Adventurers to revive after death. As a result, the Genius are often seen trying to spread death and destruction in order to acquire more empathiom.
When on Mare Tranquillitatis, Shiroe and Akatsuki saw glass-like shards rising from the sea, which are actually empathiom crystals, transformed from the snippets of hair Adventurers had cut off to symbolize their (forced) desire to return to the real world.
Minori is the first to hear about such a thing, after questioning Roe2 about her true nature. Roe2 is also mentioned to have a Sacrament-level item, an item class that never existed prior to the Catastrophe, that contains a piece of empathiom that contains a girl's voice teaching the meaning of the word "irreplaceable".
The kanji used, 共感子, transliterates to "empathy child". The name in [In Japanese, it serves as the 'proper reading' for the word], エンパシオム, has no clear translation; however, it is obviously supposed to invoke the word "empathy" and thus, feelings.
It is heavily suggested that the desire to revive is forced upon those who die. On Mare Tranquillitatis, both Shiroe and Akatsuki note that before sacrificing their memories to revive, they felt a strong urge to revive and try over again. While they can rationalize to themselves why they wanted to, there is no reason why the members of the Odysseia Knights always revive after dying, since they purposely get themselves killed in order to catch a glimpse of the old world. Furthermore, Roe2 notes that the Odysseia Knights, by dying constantly and sacrificing memories to revive, are furthering the Genius' plan, which is gathering empathiom from deceased Adventurers.