"Numele ei este Livia", I keep getting surprized by how much Romanian has inherited from Latin, both phonologically and semantically! :)
As I understand "nomen" is nominative, the subject in this case, so the order goes from what is more relevant. Literally it would be translated as "the name of her is Livia". Sorry if I don't explain myself well :/
In latin, it does not really matter which way you order the words since it is their declension that shows their use in a sentence. You could say Livia nomen ei est or Nomen Livia est ei, the meaning is the same. It is not like most modern european languages. For them, it is the position of the word in the sentence, and more specifically around the verb, that tells you what is the use of the word, if it is subject or attribute. However, there exist some practical uses in latin that you will encounter many times, such as this order.
Actually eius is the genitive for all three genders. I assume this is the Dative (ei for all genders) used to indicate possession: The name to/for her is Livia.