I would disagree that the expression is usually used to convey emphasis about a new piece of information. Rather, I would say the expression was originally used to supplement a secondary and slightly digressive piece of information to a primary statement. I.e. "Last night X and I went out to dinner at that new restaurant [primary information], where, I might add, the service was terrible [supplementary information]."
If I'm not mistaken, the current version of the expression is derived from the longer phrase "if I might [be permitted to] add...", which is the kind of pseudo-apologetic lead-in you see often if the speaker is speaking out of turn, or voicing an unpopular opinion. The modern version of that might be "In my humble opinion". I think of "I might add" in the middle of a sentence to indicate that the speaker is aware that they are digressing from the main path a little, and pre-emptively acknowledging it / apologizing for the presumption.
Of courses, these days it's used sarcastically more often than not, so that the secondary aside has the same effect as a dramatic stage whisper, hence the modern understanding of it being emphatic rather than parenthetical.