"His name is Stephanus."
Translation:Nomen ei est Stephanus.
"Ei" is the dative of "is" (he).
We already met "Mihi nomen est..." (my name is). We know that "mihi" is in dative, not nominative. Nominative for "ego" is "ego" (=I).
Ego sum = I am (nominative).
Mihi nomen = My name (dative).
It's because it means literally "To me the name is..."
And the "To me" is the dative.
So, when I say "his/her cat" I would use eius, and with nomen, it's ei?
That's just Latin idiom, which is maybe an excuse for saying that there isn't really much of a reason. The sentence is not a predicate (which would use the genitive to say that the name is "his" like a thing that he possesses), but a personal attribution, describing Stephanus's identity.
The sentence functions more like "He is called Stephanus" or, as it became in Latin's modern descendants, "He calls himself Stephanus" (e.g. lui si chiama Stefano, il s'appelle Etienne). I wonder if it accepts "He's named Stephanus" as a translation? I would argue that this possibility is correct.
I believe "is" is the nominative for "he," so no, that's not correct. I opened the SD because in English we will say "his name is," or, "her name is," using the proper possessive adjective depending on the person's gender. But in Latin it's "ei nomen est" no matter whether a feminine or masculine name follows. Is that because the possessive pronoun (ei) has to agree with the subject, nomen?
I looked "nomen" up in wiktionary and found this:
Third-declension noun (neuter, imparisyllabic non-i-stem). Sorry, the table formatting didn't work.
Case: Singular/ Plural - -
Nominative: nōmen / nōmina
Genitive: nōminis / nōminum
Dative: nōminī/ nōminibus
Accusative: nōmen / nōmina
Ablative: nōmine / nōminibus
Vocative: nōmen / nōmina
So, in conclusion nomen can either be nominative or accusative. Accusative means it's the direct object of a transitive verb, obviously not the case here, and nominative would be just as it is in Latin's descendant language, Italian. In Italian they would say, "Il suo nome e Livia" just as they would for "Marco," because the subject of the sentence is the word "name," not the person.
I hope someone who is advanced in Latin will chime in here to confirm.
This phrasing does occur in Latin. For example, the motto on Puerto Rico's coat of arms is "ioannes est eius nomen" (it's from Luke's gospel) which gets translated as "John is his name" — the emphasis is more on the name, than on the person.
These lessons are about greetings and introducing people so it makes more sense to use the dative form, to keep the focus on the people.