I don't think the right in the "The Right Honourable" means "correct", because I can't see how that makes sense in context. I considered right as a British slang intensifier that means "really", but that didn't make much sense either, because as far as I know "The Right Honourable" is a formal title that slang of lower register wouldn't be found in. So my question is: What does right add to the meaning of "honourable" here? Is there any particular reason it's there?
You're right in that you wouldn't find slang in official titles. However, according to Wiktionary, that definition is not slang, but is instead an archaic definition:
(archaic, sometimes used in titles) To a great extent or degree.
Members of the Queen's Privy Council are styled The Right Honourable for life.
In other words, it's a synonym for very. Those who are Right Honourable are honourable to a great degree, more than just The Honourable, but not quite so much as The Most Honourable.