You say that this feels like a riddle to you, but I would prefer to call it a koan. I think the best hint that we have to the nature of Bissonomy lies in the vagueness of its descriptions, as this is the sort of irony that Pratchett is famed for, and this sort of self-answering riddle has a Buddhist feel to it in turn. This also seems like a useful starting point for another reason: there are only so many potential virtues that Bissonomy can plausibly be, and the standard Western ones are already accounted for.

I suspect Bissonomy is the emotional acceptance of both that which is known to be true and that which is uncertain or unknowable. Ignorance, as we all know, is purported to be bliss, and that sounds like biss. However, Bissonomy is not the embrace of ignorance, nor does it bring bliss. Rather, it is the acceptance of that which is known to be unknowable, and it brings inner-peace.

Bissonomy and Tubso seem to be connected. Both virtues were forgotten due to their rarity. Explaining what one was should hopefully tell us something about the other. But it's even harder to say things about Tubso than about Bissonomy; the only thing that we know about Tubso is that its name is absurd. In a world where nominative determinism exists, however, this might be the only hint we need. I submit that Tubso is the virtue of absurdism, which surely has a place in Discworld. This fits with the koan framework, and establishes the desired connection between the two lost virtues; a koan's answer is absurd and hints at strange knowledge, but truly understanding a koan requires the recognition that one can never understand it fully, if at all.

Of necessity, this theory is largely speculative. We may never know the true answer to this question, and that's okay. And that, in turn, is Bissonomy.