Mother Does Not Sound Like Mather; It Sounds More like Muther

Proper spelling is an inherently flawed art that natural languages are generally unable to perfect. The problem is that to have one letter represent every possible sound might possibly require more letters than can be easily remembered, so some do double or even triple duty, and this is most particularly a feature of vowels. We have five vowels with at least ten usual forms: long and short. Furthermore, dialectical variance can give us a great deal more than that. American English is typically estimated to have thirteen vowel sounds, whereas Received Pronunciation has twenty. Remember, the actual English Alphabet only has 26 letters, and it is derived from the Latin alphabet which only has 23 letters. Having a separate letter for every vowel sound would require us to nearly double the length of the alphabet.

In A Grammar of the English Language by Samuel Johnson goes into a great amount of detail regarding how different letters are pronounced. You can see that the vowels can be used in quite a number of ways depending on the word, and some of these overlap similarly to c.

Johnson specifically writes this about the A in father:

A open is the a of the Italian, or nearly resembles it; as father, rather, congratulate, fancy, glass.

You might also note that the more Italian Mama uses the same a as in father, which may be part of the root of your child's frustration.

However, Johnson also notes that short u has a couple of sounds.

O is long, as bōne, ōbedient, corrōding; or short, as blŏck, knŏck, ŏblique, lŏll.

[…]

The short o has sometimes the sound of a close u, as son, come.

Compare son to sun, and I think you can start to get a grasp of the problem here. Some vowels have overlap, and the short o not only overlaps with u, but it also overlaps with open a and these are two distinct sounds.

The open a and the short o exemplified block and knock sounds more like the a in father, but the o in mother actually sounds more like the one in sun. Indeed, this is probably why some people call their mother Mum for short, and it also explains why some dialects slur mother into mutter.

To demonstrate the difference textually, I am going to have to resort to showing the International Phonetic Alphabet transcriptions of the word from the Cambridge Advanced Learner's Dictionary & Thesaurus's pronunciations of Mother and Father respectively.

Father "U.K. ​ /ˈfɑː.ðər/ U.S. ​ /ˈfɑː.ðɚ/"
Mother: "U.K.​ /ˈmʌð.ər/ U.S. ​ /ˈmʌð.ɚ/ "

I must admit I never did quite learn how to read I.P.A. myself, but I can observe the differences between the spellings. The takeaway here is that according to Cambridge, the ɑ following the F in father is a distinct sound from the ʌ in mother, but the ʌ in mother is shared by son, sun, and other whereas the a in father sounds more like the o in dot, otter, daughter or bother.

The exclamations ah and uh best exemplifies the difference in the sounds in mother and father respectively, because these two words isolate the pronunciation of those vowels.

Whereas O has overlap with A and U, A and U do not have quite as much overlap. I can not think of many cases where the letter A is pronounced anything like the letter U. The only instance I can think of off of the top of my head is the indefinite article, which is sometimes pronounced with a short u sound, but that is an exceptional word with two separate spellings and three separate pronunciations, so it would be a folly to use it as an example.

For the purposes of satisfying your sun son, it may suffice in this case to explain that the two words are spelled differently because they sound different. However, that will only last so long as he does not realize that some words are spelled using different letters and sound the same.

Orthography is Complicated, and It Is Especially So for Old Words

So why have overlap in the letters? Well, the art of orthography is not solely decided upon pronunciation alone. It is also decided in part upon etymological relationships in order to make it easier to recognize word families. Johnson explains this when describing overlap with the letter C for instance.

C might be omitted in the language without loss, since one of its sound might be supplied by s, and the other by k, but that it preserves to the eye the etymology of words, as face from facies, captive from captivus.

The need for matching orthography to etymology as well as pronunciation may not seem especially evident, but sometimes it helps to show that two words are related. Another example borrowed from Johnson's grammar is woman and women. Whereas woman actually sounds more or less like how it is spelled, women should be pronounced like wimen by sole right of pronunciation alone, but retaining the O helps to show that it is the same word as woman rendered into the plural form.

More importantly a language's stability is reliant upon memorization and tradition, so over the course of time the pronunciation of words can be perverted. Better record-keeping tends to mitigate against this trend, but the Gutenburg printing press was only invented around the 15th century and dictionaries only began to become popular around the 18th century and the very first audio recording is from the 19th century.

Mother is a much older than any of that, so it is not beyond cognition that it was greatly influenced by The Great Vowel Shift or some similar phenomena. Based upon what I can glean from The Online Etymology Dictionary by Douglas Harper, and Mum or Mom (or Mam) by Ashley Timms the word was originally from the baby-talk Ma, had the -ter kinship suffix slapped onto it so it was at some point pronounced like father, but somewhere along the line the pronunciation diverged. Proto-germanic used a long o based pronunciation, and the Romance languages kept the open a pronunciation.

The open A pronunciation may have been a compromise between the two, or this may have something to do in part from reduplication of Ma to make mama, which actually sounds somewhat like momma, which shortens to mom. This would gradually lengthen the noun possibly lead to a back-formation to mother.

Father does not have as clear of a path for this type of change. What is particularly noteworthy here, I think is that the P in Pa shifted to an F in Father, so a similar back formation derived from pa, papa, poppa and pop would not have made much sense, and I doubt that the vowel lengthens in the same way anyway.

Unfortunately this era of English isn't really something I know to investigate too well to confirm or deny the hypothesis. Not as many records survived and the orthography was not yet anywhere near as well settled as it is today.