H

There are such things as pronunciation guides, which are basically sets of made-up words used to stand for the letters of an alphabet to show how they should be sounded, but these show all letters, not just 'H'. To my knowledge, there is no special word for 'H' that would make it unique from all other letters, and I think you are mistaken.

The only reason why I can think you might believe 'H' is the only letter to have its own phonetic spelling may be because it seems to be discussed more than most, firstly for the reason you quote, namely that some people mistakenly "drop" the letter when pronouncing words that begin with it (eg "I'm 'ungry!"), and secondly, because there are some disagreements over the pronunciation of the letter 'H' - in some regional dialects, natives pronounce it "aitch" while others say "haitch" (the former is considered to be the correct, scholarly pronunciation).

For these reasons there is arguably more discussion over the pronunciation of this letter than any other (or at least a close second to the "zee" and "zed" variations of the letter 'Z') so you may well have seen 'H' represented phonetically more than any other.

A good question, and a very basic one that illustrates an important difference between Phonetics and Phonology (or, as it used to be called, Phonemics): They use different criteria for what's a vowel and what's a consonant.

First, an important caveat:

• This is only true of English; i.e, it's the English phoneme /h/ we're talking about.
  (This is not, for example, true of the Malay phoneme /h/.)
  Phonemes and Phonology are localized to individual languages,
  whereas Phonetics is independent of individual language systems.

What that means is that when an English speaker pronounces the words

• heat, hit, hate, hen, hat, hot,
  for instance,

they are pronouncing the phoneme strings

• /hit/, /hɪt/, /het/, /hɛn/, /hæt/, /hat/

Phonemically, and using /h/ as a Phonological consonant.

Phonological consonants are sounds that pattern in some language

1. on the borders of syllables,
2. and that are not used in those languages as syllable nuclei, like voiced vowels are in English.

The key word here is Pattern.
Phonology is all about the patterns that sounds fit into in a given language.
In English, /h/ patterns as a consonant, and that's that.

However, in pronouncing those phoneme strings
-- which represent the way speakers hear the words --
the actual /h/ sounds that the speaker says can be classified physiologically as voiceless vowels,
because a Phonetic vowel is defined by how it's pronounced,
rather than how it patterns with other sounds.

Phonetic vowels are produced by passing lung air through the open mouth
and without significant contact between any articulators.
I.e, vowels are differentiated only by the positions of the tongue and the lips.

For historical reasons, English /h/ only occurs before vowels.
It never occurs before a consonant, or at the end of a word (i.e, before Zero).
It used to occur everywhere, but those /h/'s went silent or mutated,
and are represented in English spelling as GH.
Which is why words with GH in them are so perplexing.

Now, the biggest difference between an /h/ and a following vowel is that
the vowel is voiced, whereas the /h/ is voiceless.

Further, there is not much friction necessary to distinguish an /h/ from its absence, which is the only thing it contrasts with, so all that is really needed is a transition
between voiceless and voiced occurring after the vowel has started.
Rather like the Greek concept of a "rough breathing" (Greek only had /h/ before vowels, too).

And the easiest way to accomplish this reliably and efficiently turned out to be
to pronounce /h/ with a different allophone for every vowel it preceded, like

• [i̥it], [ɪ̥ɪt], [e̥et], [ɛ̥ɛn], [æ̥æt], [ḁat]

A vowel symbol with a circle below represents a voiceless (whispered) vowel.
Any English speaker can prove this to themself easily:

• whisper eat, it, ate, ett, at, ott, holding the vowel long, to hear its voiceless sound
  then whisper each vowel, but start voicing the vowel and continue with the word.
  You'll hear an /h/ in each case, if you're an Engish native speaker.
  And, if you're paying attention, you'll notice you don't move your tongue -- only your larynx
  which means you're saying a different voiceless vowel in each case.

English has no other uses for voiceless vowels,
so they're available as allophones for /h/.
Language rarely wastes resources.