‘An hotel', with the ‘h' silent, is perfectly correct, though somewhat dated nowadays. British people of a certain class and age still do use it...it's as simple as that.
It is optional whether or not to pronounce the opening ‘h' in words in which the first syllable is unstressed...eg habitual, horrendous, hotel, historian, horrific etc and therefore whether 'a' or 'an' is used before it. The five words just listed commonly have ‘an' before them.
It is the lack of stress rather than any French provenance - as some claim - that matters. Hackney, hearse, homage and hostage all came to us from French, too, but I do not know of any evidence that British people - other than Cockneys etc - ever said 'ackney, 'earse, 'omage or 'ostage. It is probably because of such words that the relevance of French pronunciation has been doubted as regards the 'an (h)otel' usage.
Much more significant surely is the fact that all four of these words open with a stressed syllable which 'hotel' etc do not. It seems most probable, therefore, that stress-pattern is far more important than French origin in this matter. The final nail in the coffin of ‘The French Connection' is the fact that Fowler's Modern English Usage - the ‘bible' in such matters - does not even mention a French factor here. On the other hand, it does indicate the relevance of the unstressed opening.
...cont)