Why is there a word for the letter H, but not for the other letters?
In Lexico–formerly Oxford Dictionaries–for example, H = aitch, as in
‘drop one's aitches’
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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.
In response to OP's specific reference "drop one's aitches":
If I were to write this myself after hearing someone say it I would write "Drop one's h's" as spelling out the names of letters phonetically is very uncommon in writing. This phrase refers to not pronouncing the "H" sound at the beginning of some words (ex. (h)erb vs. herb).
I would follow the same pattern and type:
"cross your t's and dot your i's" as opposed "cross your tees and dot your eyes"
The second is sort of ambiguous and might be used in a humorous way.