Every church needs an icon, a graphic image for people to meditate on and identify with. It should be unique, recognizable, and embody at least some of the principles of the church.
The church of chaos is a very mixed bag but that itself is a principle. Its symbol should convey diversity, humanism, materialism, individualism.
The fingerprint fulfills these requirements. It is instantly identifiable yet it doesn't symbolize anything already. It is easy to reproduce and recognize on signs, clothing, jewelry, bumper stickers, etc. Fingerprints are everywhere but generally not visible, somewhat like spirits.
It represents individualism and diversity since fingerprints are all unique as individuals. It is humanist since it is the mark of humans. It is materialist since it occurs in the concrete world around us and not as an abstraction. There is no such thing as a human being in general and there is no such thing as a fingerprint in general; thus it represents individuality and diversity.
The association of fingerprints with the science of criminal investigation suggests the use of science and reason to find hidden truths in a complex and confusing world. This association also conveys the idea of individual responsibility since you leave your identifying mark on everything you do.
Humans touch the each other and world with their fingers so it symbolizes touch and interaction with the material world and with each other as material beings. We control machines by pushing buttons with our fingers so the fingerprint also symbolizes the interface between human and machine where we become machine like, changing our shape to push on parts of them, just like their parts push on each other.
It is the mark of individual humans in the material world. The church of chaos is for individual humans in the material world. You own your fingerprints just as you own your religion. Touch the world and be touched by it; let your fingerprint symbolize that interaction.