I would suspect that Jesus himself was the first to have the name "Jesus", but was there anyone before him with the same name? In other words, was the name completely unique when the Angel Gabriel told Mary to name her son "Jesus" or had the name been bestowed to someone else before? Today, the name is not rare (especially in some cultures), but was it so back then also?
Well, to be accurate, are you asking about the Greek name Ἰησοῦς (English transliteration: Iēsous), or the actual Anglicized (English) name, "Jesus"? Or, are you referring to his likely Hebrew/ Aramaic name, יֵשׁוּעַ (English transliteration: Yeshu'a)?
I suspect you're probably talking about his Hebrew/ Aramaic name since that was what he would have been named --- and that is the name that actually has relevance to the phrase, "...he shall save his people from their sins."
So, who was the first to be called יֵשׁוּעַ? Well, I can tell you who was the first to be named יֵשׁוּעַ in the Tanakh, but you must understand that the Tanakh does not record the name of every Israelite who has ever been born. So, there may have been people named יֵשׁוּעַ even before the first one to possess that name as recorded in the Tanakh. In any case, the first one to be named יֵשׁוּעַ in the Tanakh (if I am not mistaken) is recorded in Ezra 2:2:
There was an early Hebrew variant of the name יֵשׁוּעַ, it being יְהוֹשׁוּעַ (English transliteration: Yehoshu'a). This was the name of the one we known in English as "Joshua son of Nun." Both the names יֵשׁוּעַ and יְהוֹשׁוּעַ were transliterated into the Greek Septuagint by the proper name Ἰησοῦς.
Compare Nehemiah 8:17:
with 1 Kings 16:34:
For this reason, the KJV translators mistook "Jesus" for "Joshua" in Acts 7:45:
The passage is not referring to the Lord Jesus, but rather, Joshua son of Nun (Yehoshu'a bin Nun), the successor of Moshe, who brought the Israelites into the promised land.
As an aside, the Hebrew/ Aramaic names יֵשׁוּעַ and יְהוֹשׁוּעַ are actually quite common in the Jewish culture. For example, you can see many people by those names in the Talmud.