Moses is transliterated this way because of the way it is declined in Ancient Greek. While the root of the proper noun משה in Greek is indeed μωυση, which is roughly transliterated "moyse" or "moise", it can take the following forms based on its function in a sentence:
- Nominative: μωυσῆς (Moises): e.g. Deuteronomy 1:1: "These are the words that Moshe said". Used when Moshe is the subject of a sentence.
- Genitive: μωυσῆ (Moise): e.g. Joshua 1:1: "after the death of Moshe", and in the same pasuk, "servant of Moshe". Used to indicate possessive.
- Dative: μωυσεῖ (Moisei): e.g. Exodus 31:18: "He gave [the luchos] to Moshe"; Exodus 16:22: "and they told it to Moshe". Used when Moshe is the indirect object. (Note the subtle difference between להגיד inducing the dative versus לדבר inducing the accusative.)
- Accusative: μωυσῆν (Moisen): e.g. Exodus 2:15: "Pharaoh wished to kill Moshe". Used when Moshe is the direct object. Also all the "vayidaber Hashem el Moshe"'s are in this case.
- Vocative: μωυσῆ: e.g. Exodus 3:4: "He said 'Moshe! Moshe!', and he replied 'hineni'". Used when calling out Moshe's name.
These declensions follow particular rules, such that any noun with a certain type of stem will be declined following a particular pattern; so it is not just Moses that has a sigma added to its root to form the nominative. Some other names in the Torah that are found declined in the Septuagint are Judah/יהודה/Iουδας/Iουδα/Iουδαν, Joshua/יהושע/Ἰησοῦς/Ἰησοῦ/Ἰησοῖ/Ἰησοῦν and Sarai/שרי/σαρας/σαρα/σαραν.
The nominative form, as opposed to the root or vocative, could be considered the "primary" way to refer to the person, since it's how you form sentences having them be the subject, or the main actor.