A/an herb, for example.

Most English accents don't drop the "h" in herb, although American influence will change that.

But you can hear "an" used in phrases like "an hotel" or "an historic building", where the "h" is usually pronounced.

Most Americans don't pronounce the "h" in "herb" or "herbal" afaict. We do usually pronounce it in people's names though, like "Herb".

/heitsh/ vs /eitsh/. I think it's mostly Irish with the former.

"I think it's mostly Irish with the former."

Actually, it's a shibboleth. It used to be said in Ireland that Roman Catholic called it "haitch" and Protestants called it "aitch". "Haitch" is to be found furth of Ireland though.

And Australian, I believe.

I'm Australian, and I say 'A Hotel', and 'A Herb'. Until very recently, I'd never heard anyone ever say it differently.

GP is talking about the name of the letter H like when you're saying the ABCs. How do you pronounce it?

Pardon me. I actually pronounce it heitsh, but I don't think there's a concensus among Australians about that.

Maybe this could be used to track how some spoken language changes over time in a region or across regions.