Is there a special meaning to the class name "2-H" (2-ふ)? Sayonara Zetsubou Sensei is focused on the students of the class 2-H. Is there a hidden meaning to that class name or is it just a common class name in Japan?

I recently saw some other, possibly related anime, also with students in class 2-H, but I can't remember which one.
I'm not sure where you got the information that the students are in class 2-H. Wikipedia lists Nozomu's class as 2-へ (2-He). Here, the number indicates the year, while the character indicates what class in the year they are in. Since へ is the 6th character in Iroha order, this would correspond to class 2-F. Incidentally, 2-ふ would be the 32nd class in year 2, which is far more than most Japanese high schools would have.

There's nothing really unusual about having 6 or more classes in a given year. It is a bit unusual that they're labeled in Iroha order, which is pretty old fashioned. It's more common (in my experience) for classes to be labeled with letters in the Roman alphabet or with numbers. I don't think there's any deep meaning to using this though; it just conjures a sort of old-fashioned feeling.

I can't think of any hidden meaning to 2-へ, but I have not read to the end of the manga so there could be something revealed at the end. If there is such a meaning, though, it isn't obvious and it certainly doesn't apply to other anime which have this same class name. The name is a pretty normal one, if old fashioned.