Is there a rule for the pronunciation of the "H" at the beginning of a word?

Nope. Probably why English is the most if not one of the most difficult languages to learn.

Great question. I don’t know of any rule in English regarding the H at the beginning of a word except that most of the time we do pronounce the H, but there are many exceptions. You just need to memorize as far as I know. It will be interesting to see if anyone knows a rule. Possibly when to pronounce it has to do with the language of origin. In America we tend to come close to the pronunciation if the country if origin. I assume herb is a French word? Not sure. Do they make the H silent? While a word like halt is from German I would think.

Well, both of the silent examples start with HO. Don’t know if that has anything to do with it. Originally it’s possible that they did use the H sound, and we just morphed it away through laziness.

@Dutchess_III Herb, as in spices and herbs, typically is pronounced with a silent h. I know some people pronounce the h.

Heir is another.

Hope, hole, hop. It can’t be just the h-o beginning.
It matters especially since the ones where you don’t pronounce it call for an “an” as a preceding article instead of just an “a”. The only other silent-h word I can think of is “heir”.
(In my opinion, human or any other non-silent h word should be preceded by an “a” article.)
ThisThis website explains it a little.
This talks about it, too. It looks as though word pronunciations simply evolved.

There are other words where the H is silent if you include words where the h is not the first letter.

I agree with @kritiper and no, I don’t think there is a rule. It’s just something you learn by ear (I think but I’m not a linguist). Given how many English speaking people drop their aitches, I’d guess many native English speakers a. have bad pronunciation and/or b. can’t work it out.

@JLeslie In NZ the word herb is always pronounced with an h, I thought the silent h was just a US thing, it’s always rather puzzled me because it sounds so weird without the h.

@Adagio It might very well be an American thing. I think in my first answer I was careful to say in America we tend to come close to the original pronunciation, but in America we also change a lot of things in language over time. My guess was herb is from a language where the h is silent. I’m not sure though. In words like herbivore we say the h.

I’ve heard Gordon Ramsey say steak filet as fill-et as though it was spelled fillette. We usually pronounce words that end in et that are from the French similar to the French; Fil-ā (I hope that’s the right a. Should be what rhymes with day). Ballet would be another word that is similar.

@Adagio in the UK we always pronounce the H in herb too. Well, depending on your dialect, I think they don’t pronounce H at all in the south-west of England, in any word.

Thanks for your responses, everyone! Seems like this will stay an unsolved mystery ;)