The name Josiah in the Bible
There's really only one person in the Bible named Josiah, and he is a son of Zephaniah. It seems he ran a kind of reception center for the returnees from exile (Zechariah 6:10). The prophet Zechariah also mentions a Hen, son of Zephaniah, who may be a brother of Josiah or, as most source suggest, he is the same person by another name (Zechariah 6:14).
The most famous Josiah is actually called יאשיהו (Josiahu). He is the son and successor of king Amon of Judah and his mother is Jedidah (1 Kings 13:2). King Josiah is remembered as a good king, who destroyed much of the centers of pagan worship, and cleaned up the temple in Jerusalem. King Josiah died in battle against Egypt on the plain of Megiddo; this battle may be the one after which visions of the final battle of Armageddon were modeled.
King Josiah is also an ancestor of Jesus (Matthew 1:10). His name in Greek is Ιωσιας; Josias.
Etymology and meaning of the name Josiah
The name Josiah consists of two elements. The final part is יה (Yah) = יהו (Yahu) = יו (Yu), which in turn are abbreviated forms of the Tetragrammaton; the name of the Lord: YHWH.
The origin of the first part of the name Josiah is unclear. To start with, it looks exactly like the verb יאש (ya'ash), meaning to despair:
Thus the name Josiah also carries the unmistakable meaning of The Despair Of Yahweh.
BDB Theological Dictionary dictates that the name Josiah derives from the verb אשה ('shh), which is unused in the Bible but which probably means to support. Its derivative אשיה ('oshya) means buttress or pillar and occurs only in Jeremiah 50:15. Hence BDB reads Yah Supporteth.
This verb, however, is strikingly similar to the word אשה ('ishsha), meaning woman or female, which comes from the root group אנש ('nsh), having meanings like to be sick, weak or social (see the name Enosh):
And what to think of the similar noun אשה ('ishsheh), meaning a fire offering (Leviticus 1:9). This word probably comes from the word אש ('esh), meaning fire:
Alfred Jones (Dictionary of Old Testament Proper Names) assumes this mystery root indeed once existed, and liberally reads Given Of The Lord as meaning of the name Josiah.
Younger sources point at a possible meaning of Yahweh Heals (NOBSE Study Bible Name List) or Whom Jehovah Heals (Spiros Zodhiates—The Complete Word Study Dictionary of the New Testament). These seem to suppose that the name Josiah was imported from another language because the Hebrew verb to heal is רפא (rapa—hence the names Riphath and Rapha).
Since we have no sound recordings from the olden days, it can't be said with any certainty what the name Josiah sounded like to a Hebrew audience. Perhaps the noun for fire came from a verb meaning "to make fire," and the name Josiah reflected God's creating fire: Fire Made By YHWH