It just occurred to me that this construction is very peculiar.
Pronoun: I had better get going. NP: The cat had better be home. Expletive: There had better be food on the table. Gerundive C: Eating cake had better be good. Wh-C: What he saw had better be worth seeing. Infinitive C: ?To fish had better calm my nerves. Relative C: *That she kissed the frog had better amuse me.
The subject has to be a NP or a pronoun and can't be a relative clause. This may be because relative clauses are abstract and can't be the things or actions someone would expect to have done or been something. Infinitive clauses should be grammatical, but I'm not so sure. The verb that follows "had better" being infinitive is, I suspect, due to the subjunctive.
I can't think of any other example where it would be grammatical to substitute the word "better" with any other comparative (adj/adv), noun, or verb ("better" can be a noun and infinitive verb).
What part of speech is "better" in the construction? Are there any other constructions quite like it? Is this an example of a fossilized construction?