‘An hotel', with the ‘h' silent, is perfectly correct, though somewhat dated nowadays. However, British people of a certain class and age do still 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 probably the lack of opening stress rather than any French provenance - as some claim - that matters, although opening letters ‘h' in French itself are not spoken . 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 - hackney, hearse, homage and hostage - open with a stressed syllable which the other four - hotel, habitual, historian and horrific - do not. It seems most probable, therefore, that stress-pattern is more important than French origin in this matter. Fowler's Modern English Usage does not even mention a French factor here, though it does indicate the relevance of the unstressed opening.
...cont)