Why is Hermione pronounced like her-MY-on-ne in English? Does it follow the rules? It doesn’t seem phonetic, and the Greek is probably different.

It follows the rules alright. They’re just rules that have nothing to do with the original Greek.

Traditional English pronunciation of Latin – Wikipedia

In the middle of a word, a vowel followed by more than one consonant is short, as in Hermippe /hərˈmɪpiː/ hər-MIP-ee, while a vowel with no following consonant is long.

Hence, Hĕrmīone. (Long and short as in Modern English spelling: long i = [aj].)

Endings: … The first class consists of vowels alone, i.e. -a, -e, -æ, -i, -o, -u, -y. In this class, the vowels are generally long, but -a is always /ə/.

Hence, Hĕrmīonē.

Latin stress is predictable. It falls on the penultimate syllable when that is “heavy“, and on the antepenultimate syllable when the penult is “light”. … A syllable is “light” if it ends in a single short vowel.

Hence, Hĕr-mī´-o-nē.

However, when a vowel is followed by a single consonant (or by a cluster of p, t, c/k plus l, r) and then another vowel, it gets more complicated.

• If the syllable is unstressed, it is open, and the vowel is often reduced to schwa.

Hence, Hĕr-mī´-ŏ–nē, [hɜɹˈmajəniː]. As opposed to the Ancient Greek [hermiónɛː], or the Modern Greek [ermiˈoni].