We need a character for h with diaeresis. Well, there already is one (ḧ -- you might not have the right font to read that) but it's not in common use. Actually, it's not in use at all. Wikipedia acknowledges its existence but does not have an article nor mention any use for it. I would suggest it be used like ä ë ï ö and ü are used in English: namely, to indicate the start of a new syllable where it might be missed, like in naïve or coördinate.
There are many places where one could use an h-diaeresis! Take The Great Gatsby. There's a character named Wolfsheim. That's not pronounced WULF-shaim, it's pronounced WULFS-haim. Other words which could benefit from an h-diaeresis include the following: lighthearted, lightheaded, butthead, misheard, fathead, shepherd, lefthanded, and, of course, asshat. I've already started using h-diaeresis in my homework, but it's not that often that I get to use it. Maybe someone will notice eventually.
