What does the "right" in the "The Right Honourable" mean? Why is it there?

Solution 1:

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.

Solution 2:

Right Reverend is a similar title, applied formally to Anglican bishops where ordinary priests are styled Reverend (but not to archbishops, who are styled Most Reverend). The Orthodox churches use this wording too, so it's not purely British.