Hour starts with a quiet h — it sounds just like our. Rhyme has a quiet h too.
The rule
H is silent in a small set of French-derived words — hour, honest, honour, heir — and in the Greek rh- pattern: rhyme, rhythm, rhino.
What is really going on
The h-dropping set comes from French, where initial h was not pronounced. English restored the h in most such words (history, hospital, hotel) but not in hour, honest, honour and heir. The rh- group is Greek, where rh transliterates rho with a breathing mark.
Position in the word
Beginning of a word — A small French-derived set. For example: hour, honest, honour, heir.
After a consonant — After r, in Greek words. For example: rhyme, rhythm, rhino, rhubarb.
h restored: history, hospital, hotel, human, horrible. Most French h-words had the h put back into pronunciation.
Contrast pairs
hour versus our — Homophones.
honest versus honey — One has a silent h, the other does not.
Teacher note
Teach hour, honest and honour as a three-word list — they are frequent and they do not generalise. Rh is better taught with the Greek origin group, alongside ph, ch as /k/ and y as /ɪ/.