"Had better" — what is the meaning of this grammatically?

I'm interested to know why we use had better for recommendation. Technically we're speaking of an action that hasn't yet occurred. Like he had better leave a tip means he hasn't yet left a tip, but I suggest he do.

Why do we use the past tense had if it has nothing to do with the past? Why do we not use the subjunctive here? (He have better leave a tip.)

The NOAD reports that the meaning of had better do something is

• would find it wiser to do something (you had better be careful)
• ought to do something

It reports also this note about the usage:

In the verb phrase had better do something, the word had acts like an auxiliary verb; in informal spoken contexts, it is often dropped, as in you better not come tonight. In writing, the had may be contracted to ’d (you’d better call), but it should not be dropped altogether (not you better call).

"You had better..." seems to be an expression (idiom) that functions as a strong suggestion to solve a problem. It seems to be even stronger than "You should..." Examples:

• You had better see the doctor about your cough.
• You had better study hard for your entrance exam.

To explain the grammar, I'd say it is such a strong suggestion phrase that it suggests something that the person should have already completed in the past but is just starting to do. Clearer grammar might be "You better have finished the report!" instead of "You had better finish the report!"

Look at this sentence

He had better leave a tip.

It's actually used to say someone should do something. But it makes sense to tell that the real form of the sentence was:

If he had left a tip, it would be better.

It means he has not done the work, but still he has the opportunity to do it.