The fact that consonant names are all more or less built on the same model is a rather modern phenomenon in the history of the alphabetic letters' names.
In the past, all these letter were named after the thing they represented.
Have you noticed for instance that if you rotate the letter A by 180°, it looks like a cow head? Well that's what the alf letter is supposed to represent in the original Phoenician alphabet: an ox (named alf => alef in Hebrew, alpha in Greek).
So Phoenicians kids would probably not say "aaaah" but "alf" (alf is actually not a vowel but the "stop" - there are no vowels in the Phoenician alphabet). Greek kids in turn would probably say "alpha", but one can already notice that we've passed from the name of the thing to the name of the letter itself because the Greek for ox is not alpha but boos (βοῦς).
Therefore the real reason why letter names have changed over time is actually that the original names are only meaningful in the language of their inventors and that they are thus harder to memorise by later adopters and therefore more likely to be simplified.
For the record, the simplified path from the "invented" Phoenician alphabet to today's "English" (Latin or actually post-Latin) alphabet is as follows.
- Phoenician Alphabet mainly local invention with Egyptian hieroglyphs inspiration (11/10th century BC) .
- Supplanting linear A and B in Greece in the wake of the intense trade between the Levant and archaic Greek kingdoms (9th century BC). In these times Greek was still written from right to left or even boustrophedon. In the process, the Greeks promoted a few unused consonants to fully functional vowels.
- Imported in Italy to write down Latin, Oscan, Etruscan and other early Italic languages (8th century BC).
- Supplanting futhorc runes as a result of the Christianisation of the heptarchic kingdoms (7th century AD).
Just a few more letters:
The B (also rotated, this a general rule of hand writing), is supposed to be a birds eye view of a house (=> in many Semitic languages : beth).
O: the lower case Greek omicron used to have a dot in the middle in archaic scripts. And the upper case still retains an horizontal bar. No wonder: that's supposed to represent an eye.
The M (pronounced mem in Pḧoenocian) resembles a lot the Egyptian hieroglyph representing water ripples (mem). Explanation of "em" ?
As for the H (pronounced "aka" in Latin and Italian, "ah" in German, "ash" in French), it is still close to its initial pronunciation: a wall (het => heth in Hebrew, ḥā in Arabic). It cannot follow the normal rule "bee", "cee" "dee" because it's aspired or silent (as other posters have convincingly shown). That would make it "hee". It does not sound right! Does it?