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Mouse and louse were similarly mūs /muːs/ and lūse /luːs/ (like moose and loose) in Old English, but their plurals were mȳs /myːs/ and lȳs /lyːs/, respectively.

1. hūsa- > hūsō // mūs > mūsiz (pre-English/Proto-Germanic) 2. hūsa- > hūso // mūs > mȳsi(z) (pre-English, i-mutation, final syllables weakened) 3. hūs > hūs // mūs > mȳs (~ Old English, loss of final syllables) 4. hūs > hūs // mūs > mīs (late Old English, unrounding of /y/ to /i/) 5. həus > həus(en/es) // məus > məis (Middle/Early Modern English, Great Vowel Shift diphthongisation, house starts getting an explicit plural) 6. haʊs ⟨house⟩ > haʊzəs ⟨houses⟩ // maʊs ⟨mouse⟩ > maɪs ⟨mice⟩ (Modern English, diphthongisation complete, house has regular plural)

Mouse and louse were similarly mūs /muːs/ and lūs /luːs/ (like moose and loose) in Old English, but their plurals were mȳs /myːs/ and lȳs /lyːs/, respectively.

1. hūsa- > hūsō // mūs > mūsiz (pre-English/Proto-Germanic: starting point) 2. hūsa- > hūso // mūs > mȳsi(z) (pre-English: i-mutation, final syllables weakened) 3. hūs > hūs // mūs > mȳs (~ Old English: loss of final syllables) 4. hūs > hūs // mūs > mīs (late Old English: unrounding of /y/ to /i/) 5. həus > həus(en/es) // məus > məis (Middle/Early Modern English: Great Vowel Shift diphthongisation of /uː/ and /iː/ to /əu/ and /əi/; house starts getting an explicit plural) 6. haʊs ⟨house⟩ > haʊzəs ⟨houses⟩ // maʊs ⟨mouse⟩ > maɪs ⟨mice⟩ (Modern English: diphthongisation continued to /aʊ, aɪ/; alternative plurals of house disappear, leaving just one, regular plural)

Let’s start with house(s). The reason why the plural of house is houses is that that is the regular pattern. That simple. In earlier stages of English, house had different plurals; but it was regularised to fit in with the most basic pattern of adding /z/. If we go back to Old English, the word was hūs (pronounced /huːs/, like ‘hooce’ in Modern English), and the plural was also hūs.

It does sometimes happen that a regular word becomes irregular if there is enough pressure (for example, dive has developed the past tense dove in American English because of the similarity to drove, strove, throve), but it is much, much more common that unpredictable, irregular forms are lost in favour of regular forms—so if anything were to happen in future to make the three words in this question the same, the expected development would be that mouse and louse become mouses and louses.

Let’s start with house(s). The reason why the plural of house is houses is that that ending is the regular pattern.¹ That simple. In earlier stages of English, house had different plurals; but it was regularised to fit in with the most basic pattern of adding /z/. If we go back to Old English, the word was hūs (pronounced /huːs/, like ‘hooce’ in Modern English), and the plural was also hūs.

It does sometimes happen that a regular word becomes irregular if there is enough pressure (for example, dive has developed the past tense dove in American English because of the similarity to drove, strove, throve), but it is much, much more common that unpredictable, irregular forms are lost in favour of regular forms—so if anything were to happen in future to make the three words in this question the same, the expected development would be that mouse and louse become mouses and louses.

^{1 The ending is regular; the form as a whole is slightly irregular, since the stem-final consonant /s/ is most commonly (though not consistently) voiced before the plural ending. This is a pattern found with many words that end in unvoiced fricatives (/f θ s/); cf. mouth /maʊθ/ ~ mouths /maʊðz/, life /laɪf/ ~ lives /laivz/. For all three consonants, though, the plural voicing is sporadic and only happens sometimes—there is no rule.}

Well, hūs is a neuter noun in Old English, while both mūs and lūs are feminine nouns. V0ight’s answer has already mentioned i-mutation (also known as i-affection), which is the historical cause of the different vowel in the plural of the latter two words. Historically, the plural ended in -i (like ‘-ee’), and this high front vowel caused the preceding vowel to assimilate, to become more ‘ee-like’. At some point in time, before the stage we know as Old English, the final /iː/ was lost, but the change it had caused in the preceding vowel remained.

This ‘ee-like’ plural ending was only used in the masculine and feminine genders, though; not in the neuter. In the neuter, there were various other ways of forming a plural, including not adding an ending at all. We can see from various comparative evidence that an earlier form of hūs also had an extra syllable lost in Old English, but in hūs, that extra syllable had an /a/, not an /i/ (Proto-Germanic *hūsa- was an a-stem, so its plural would have been *hūsō in pre-English times). Since there was no /i/, there was nothing to cause the i-mutation.

So if we go historically, the development was something like this, giving singular > plural pairs for ‘house’ and ‘mouse’:

Well, hūs is a neuter noun in Old English, while both mūs and lūs are feminine nouns. V0ight’s answer has already mentioned i-mutation (also known as i-affection), which is the historical cause of the different vowel in the plural of the latter two words. Historically, in Proto-Germanic, the plural ended in -iz (pronounced much like ‘-eez’ would be in English), and the high front vowel /i/ in that ending caused the preceding vowel to assimilate, to become more ‘ee-like’. And an /u/ that becomes more ‘ee-like’ almost always becomes /y/, as indeed it did in English. At some pre-English point in time, this final syllable was lost, but the change it had caused in the preceding vowel remained.

But this ‘ee-like’ plural ending was only used in the masculine and feminine genders; not in the neuter. In the neuter, there were various other ways of forming a plural, including not adding an ending at all. We can see from various comparative evidence that an earlier form of hūs also had an extra syllable lost by the time of Old English, but in hūs, the vowel in that extra syllable was an /a/, not an /i/ (Proto-Germanic *hūsa- was an a-stem, so its plural would have been *hūsō). Since there was no /i/, there was nothing to cause the i-mutation and change the /ū/ to /ȳ/.

So if we take it chronologically, starting from the Proto-Germanic stage, the development house and mouse went through went something like this (giving singular > plural pairs):