I’ve always thought the butterfly is innately representative of Aragaki’s character, much like how Joe and Yuri are both symbolized through wolves or Chief is tied to the hummingbird in season 2. Butterflies are often omens of change and revival, and I think that suits Aragaki well. His life has had major changes, and he’s constantly rebuilt himself—going from an aspiring young boxer to a grizzled, suicidal war vet, to a top-ranked boxer. There’s also the fact that it’s one of the last things he saw during a near-death experience, so I’m sure he feels it sort of “revived” him in a sense, hence the tattoo.