The e is easier to explain - it is used before people's names when you address them directly, such as in English "Hey (name)".
The 'o / 'O is used in two different places. In this case, it starts a sentence without a verb. It is used before definite nouns (the, these, those, my etc. + a noun) or a proper noun like the name of a person or a place. If it is an indefinite noun (a, some) then He is used instead of 'O. The word 'O has no English translation.
In addition to the solid info that Kelii provided 'O at the beginning of the sentence signals a particular sentence type ('Aike 'O) an equational sentence. That means the two things mentioned (the po'o/head and the piko/body) are equal. So you can reverse the two items and says the same thing: 'O Ka'iulani kona inoa (Ka'iulani is her name) or Kona inoa 'o Ka'iulani (Her name is Ka'iulani).
This sentence structure is use to equate two nouns/pronouns. Because of the nature of this type of sentence, that it has no verb, the two halves can be swapped and have pretty much the same meaning. However, regardless of what you put first, it needs to start with ‘O.
Just a word of caution, sentences that start with He (equating two nouns where one is indefinite (a/an) then those two halves cannot be swapped except in the negative.
I don't understand your statement about being fair to someone typing it (? - typing what? the translation?). Nevertheless, as Kelii explained, the is an 'Aike O or equational type of sentence and must start with 'O. However, proper nouns and pronouns must be proceded by 'o, so one of the choices will have an 'o at the beginning and other before the name at the end. See both forms in my reply above.