Andrew has a new post up describing his new way to think about Koide relations.
I'd say it's characterized by two ideas. First, the Higgs boson is somehow bundled with the W and Z bosons, and so mass generation is regarded as intimately linked with the weak force. Second, Koide relations arise from a sort of equilibrium of weak-force transitions between the fermions in question.
There's a lot to be said about the compatibility (or incompatibility) of these ideas with more commonplace theoretical notions, like seesaw or Froggatt-Nielsen, and with standard notions of the weak force, the Higgs mechanism, QFT, etc; and also much to be said regarding how these ideas could be given sufficient specificity to become the basis of a mathematical theory. Much to think about!