If the word once had a silent h (because it was borrowed from French), use an. Otherwise (f'rinstance if it's borrowed from Greek), use a. Therefore it would be an hotel but a hypothesis.Not so. I looked it up. Nowadays, of course, the rule is to use a wherever [h] is pronounced (a hotel), and an wherever h is silent (an honour), and very sensible the rule is too. But if one did want to use an, one should properly do so with words longer than 'about three syllables' and which have an unstressed initial syllable. Hypothesis, for instance.
And it turns out not to be a stupid left-over-from-history rule either: it really is easier to say an when the syllable is not stressed, because it takes too much effort to stop after a and start again on the relatively weak [h] sound in an unstressed syllable.