The letter /h/ just happens to be the letter that we use in digraphs in modern English. Digraphs are a group of two letters that represent a single phoneme. That's why it's used to modify other letters to create a single sound--as we lost our special characters in English, such as the yogh, the aesc and the thorn, we replaced them largely with digraphs. The letter h was used because of it's relative evanescence in spoken language. Some dialects pronounced it, some didn't, and these things have switched off over the years, so h was a logical choice because it was there, but didn't seem to have a whole lot of use. In Polish, they use a -z- for this purpose, and in Hungarian as well.

You should try posting this to /r/linguistics.

In English, wh was once pronounced [ʍ] (roughly equivalent to /hw/) and gh was pronounced [x], but both of those sounds dropped out of use in RP and standard American English (although the wh sound is preserved in Scottish and Irish English).

In French and Spanish, while the h sound has now become silent, that has not always been the case. French actually lost the h sound in two stages, once during the Early Middle Ages and again around the late 13th century, which is why French makes the distinction between h aspiré and h non-aspiré.

tl;dr, the letter h, whether on its own or in combination with other letters, often represents sounds that have later become extinct in a language.