According to Michael Erler, professor of philology at the universtity of Würzburg, translating eudaimonia as happiness is not quite accurate. A more accurate translation would be flourishing.
Just like the Latin term 'augustus', the Greek term eudaimonia carried strong religious connotations which are already apparent from its etymology. Daimon translates to deity or divinity and eudaimonia is when a divinity is well disposed towards you, so the term comes pretty close to modern 'blessedness'.
The closest ancient Greek equivalent to 'personal happiness' would be 'eupraxia', a noun derived from the quite common expression 'eu prattein' meaning 'to be doing well in life' or 'to be successful'.
In modern Greek the general term for 'personal happiness' is 'eutychia' which in ancient times was more confined to the sense of 'good luck' which is modern Greek is simply tyche. In modern German, the term 'Glück' (related to English 'luck') is the standard term used both for the sense of 'personal happiness' and for the sense of 'good luck'.
Those in ancient Greece who taught people how to be 'happy' in the modern sense of 'doing well in life' were actually the sophists who professed to teach people how to be successful in both their public and private affairs for a price. It was not the philosophers. The philosophers were generally skeptical of such definitions and approaches to success and focused on teaching practical ethics.
Erler writes:
For Epicurus is convinced, as he says at the end of the Letter to Menoeceus as well, that whoever obeys his advice day and night will achieve freedom of pain with regard to the body, i. e. aponia, and freedom of pain with regard to the mind, i. e. ataraxia. In short, they will be happy. Happiness is feasible! That is Epicurus’ message, which might sound strange to many because of our different understanding of ‘happiness’. It is important to keep in mind that Greek eudaimonia (or happiness) has a meaning which differs from our modern understanding of happiness. For ‘happiness’ today is commonly regarded as a subjective mood, a feeling that can change from
day to day and can be influenced by new situations. For the ancient Greeks,
however, eudaimonia, which is usually translated by ‘happiness’ but which
rather should be translated by something like ‘human flourishing’, was not an emotional state, but rather about whether a human being had attained virtue and excellence, achieved his aims, and truly made the most of his life.
This understanding is well illustrated by Herodotus in his Histories where he
tells the story of Solon making an important point about human happiness in a conversation with Croesus: Any human life according to him is filled with change, so that a person’s life cannot be evaluated as ‘eudemonic’ properly until he or she has died. This remark follows from the understanding of happiness as a fulfilled life, which is the reason why Epicurus – and Plato for instance – can claim that he offered doctrines helping to become happy by helping to live a good and happy life. Only if eudaimonia is understood in terms of ‘fulfilled life’ does it really make sense to say, as Epicurus does, that one can teach someone to become 'happy'.
I've brought up "well-being" = eudaimonia so much even I'm tired of my rants!
