“Uni” is a prefix meaning “One”; while “verse” derives from Old French root word meaning “to turn”. Years ago I assumed a different meaning. The “one verse” to me was the “One Truth.” There weren’t religious connotations to it, I just thought that the root of veritas was the same root as verse, but alas it’s not. So my youthful assumption that it meant “One Truth” was dispelled the day I looked it up in my OED a few years back.
Instead if you read through the roots of associated words and their derivatives, you get the sense that it could mean one fate, one turn, one line, or — one verse.
At times I still muse about one truth and prefer my original interpretation to this day. Though untrue it’s a useful and beautiful delusion to be pondered now and then; and once pondered then shelved once more in the depths of memory with other fantasies.