🔼The name Jose in the Bible
There's only one Jose in the Bible. He is mentioned in the Lucan genealogy of Jesus, as a son of Eliezer, father of Er and great-great-great-great-great-great-grandfather of Zerubbabel, the leader of the second and most substantial wave of returnees from exile (Luke 3:29).
For some mysterious reason, no version of the English Bible apart from the good old King James Version and the Young translation, simply prints Jose. The two big modern ones, the New International Version and the New American Standard, both speak of Joshua. The American Standard Version has Jesus, and Darby has Joses (The form Ιωση occurs in Matthew 27:56, Mark 6:3, 15:40 and 15:47 as the genitive form of Joses).
Our name Jose is probably an truncated form of the name Ιωσης (Joses), which in turn comes from the Hebrew name Joseph (in Hebrew: יוסף, and transliterated into Greek: Ιωσηφ), but many people insist it's a form of the name Ιησους (Jesus), which in turn comes from the Hebrew name Joshua (in Hebrew: יהושע, and transliterated into Greek: Ιησους), which is obviously a completely different name.
🔼Etymology of the name Jose
The name Jose is one of a few Greek names that are derived from the Hebrew name Joseph, and it became one of the most popular names in the modern Spanish speaking world. The name Joseph comes from the verb יסף (yasap), meaning to add, increase, or repeat:
As also affirmed by Spiros Zodhiates (The Complete Word Study Dictionary — New Testament), the name Jose means the same as Joseph, which Zodhiates translates with May God Add. Note, however, that the "God" part is implied and not actually incorporated in the name Joseph.
For a meaning of the name Joseph, NOBSE Study Bible Name List reads Increaser, or slightly more descriptive and assume that the "He" is YHWH: May He (Yahweh) Add. Jones' Dictionary of Old Testament Proper Names has He Shall Add. And BDB Theological Dictionary proposes He Adds, Increases.