The name "Jesus" has a long, long history. The origin of this name is the Hebrew name יהושע (yehoshu'a, Strong's #3091 [Latinized as Joshua]), which means "Yahweh saves." This Hebrew name is first used Exodus 17:9 where we are introduced to Yehoshu'a Ben Nun.
When this Hebrew name was transliterated in the Greek Septuagint (2,000 year old Greek translation of the Hebrew Bible) it was written as Ἰησοῦς (iesous). The Greek alphabet had no "Y" sound, so it used the "I" sound. The Greek alphabet has no consonant "H," or equivalent, so this sound is dropped. The Greek alphabet also had no "Sh" sound, so it used the "S" sound. Then, Greek male names end with "s," so the "s" was added. And this is how יהושע (yehoshu'a) became Ἰησοῦς (iesous) in the Greek.
It is common for names to shift and evolve when transferred from one culture to another. For instance, the German name Ludwig is Louis in France and the Spanish name Juan is John in English speaking countries. The Hebrew name יהושע (yehoshu'a, Strong's #3091) is ישוע (yeshu'a, Strong's #3442 [Latinized as Jeshua]) in the Aramaic language, such as we see in Ezra 2:2. When this Aramaic name was transliterated in the Greek Septuagint, they used the same method as stated above and it comes out to Ἰησοῦς (iesous), the same as it did for yehoshu'a.
When we come to the New Testament period we find that the name of the Messiah is Ἰησοῦς (iesous) in the Greek New Testament, but we find that it is ישוע (yeshu'a) in the Aramaic New Testament. When the Greek New Testament was translated into Latin in the 4th Century this name was written as Iesus, an exact match to the Greek that it came from. The Latin letter "I" split into two letters, "I" and "J." Originally this was two different ways of writing the same letter. So the Iesus became Jesus, but they were both pronounced the same way. Years later, some cultures began using the "I" for the vowel sound and "J" for a "Y" sound. It was not until around 1500 AD that the letter J took on the "dg" sound we are familiar with today.
So, the modern name "Jesus" comes from the Latin Iesus, which comes from the Greek Iesous, which comes from the Aramaic Yeshu'a and the Hebrew Yehoshu'a.
