I understand that there is an ellipsis in "It better be", representing "It had better be", but what sort of ellipsis is this? Wiki suggests a number of different examples but none seem to really match in the eyes of this rank amateur.
It's not an ellipsis, it's the old subjunctive mood, its present tense form (I be, thou beest, he be), used in the optative sense to express a wish or hope. For example, "God save the Queen" uses the same grammatical form ("I hope God saves the Queen")
"It had better be" technically is subjunctive+preterite
My assumption matches Jlawler's (that phonology is the culprit). Further, I have assumed that the examples cited by StoneyB ("I got rhythm" and "You been on my mind") are the same phenomenon (but perhaps not "You done done", which I would analyze differently).
As far as what grammatical form we're left with after the /d/ is lost, and we have "It better be", my assumption is that "better" then becomes a modal. That would explain why we don't say *"Does he better be careful?". Of course, we also don't say *"Better he be careful?", but I would attribute this (perhaps) to the fact that (perhaps) it's a newer modal whose paradigm hasn't been fully formed yet. Also, some of the other modals seem to be somewhat deficient in this regard as well; "may" can be used for probability or for permission, but I do not recall hearing or reading "May I be there?" for probability -- only for permission.
I doubt the phonology explanation. My most natural pronunciation of "It better be good" is [ɪʔbɛɾɹbiɡʊd]. If there was a phonemic "had" in there, how could it have disappeared? The first two things to happen in the reduction to a casual speech form would be loss of the "h" of "had" and then flapping the "t" of "it". But once that "t" is flapped, there is no longer any source for the glottal stop that you wind up with.