To summarise what previous posters have said:
Esme (acute accent over second e) is French. It is a past participle, used as an adjective. It means 'esteemed' or 'loved'. In French, it is pronounced 'Ez-meh' or Ez-May'. It was first used in the British Isles in Scotland, in the 16th century, as a male name (for a half-French nobleman who was a 'favourite' of King James VI and I). Hence the male-adjective spelling.
French participles/adjectives agree with the gender and number of the word they are describing. So the female version would be Esmee (acute accent on second e). Pronunciation also Ez-meh or Ez-May.
Esme has nothing to do with the name Esmeralda, which derives from the Spanish word for emerald.
Over the years, some people in the UK have used Esme (acute accent over second e) as a unisex name, even though it is grammatically male.
Some people in the UK say 'Esmee', but following the rules of the French language, technically they are wrong. (Just as the previous poster said about Nestle/Nesstil. ) However, in spite of Mumsnet, there is no language police. If people choose to follow a technically incorrect pronunciation, no-one is actually going to stop them. It may be - who knows - that in the future the technically-incorrect pronunciation of a French name will eventually become a new and independent English language name, like, as previous posters have said: Aimee (acute accent on first e) and Amy.
If I were naming a daughter Esme or Esmee, however, I would pronounce it 'Ez-meh' or 'Ez-may'.