The difference between sibi and ei in this case is that sibi refers to the subject of the sentence – that would be nomen in this case, so would make no sense – while ei refers to some other person or thing, usually previously mentioned.
Note that mihi and tibi do not have the same property. They just refer to the speaker or the addressee, no matter whether they're the subject or not.
Since the caption of this question is "possessive pronouns," I must mention that these are all not possessive pronouns, but possessive pronouns (and possessive adjectives – they are the same form in Latin) do exist and, lo and behold, exhibit the same phenomenon:
- mine/my = meus
- thine/thy = tuus
- his/her/s = eius (non-subject) / suus (subject)
Grammatically it would be fine to say Nomen eius est Stephanus or Nomen meum est Ionas, it just appears that this is not how Romans typically told their own or someone else's name.
Aulus Gellius (5, 29) wrote:
Duae istae in loquendo figurae notae satis usitataeque sunt: «mihi nomen est Iulius» et «mihi nomen est Iulio»
These two expressions are known and common enough in speech: “mihi nomen est Iulius” and “mihi nomen est Iulio.”
He then goes on to mention a third form he discovered in the Annals of Piso (lost to us), which, however, must have been rare.
So there you have it: You either use the dative or nominative form of the name with the dative of the named person (or a pronoun) + esse.