So everybody wants to know where I got this name for my son from. Many people never hear of it before so here is a quick history lesson.
Yusha (pronounced You-Shaa) is the Arabic equivalent of Joshua.
According to Ibn Katheer in Bidayah Wa Nihayah, his lineage is as follows, he is Yusha the Son of Noon, the son of Afra'yeel, the son of Yusuf, the son of Yaqub, the son of Ishaaq, the son of Ibraheem (Peace be upon them all). So Yusha is the great-grandson of prophet Yusuf (AS) and out of six of his forefathers, four were prophets.
Yusha was the prophet Musa's (AS) companion on his journey to meet Khidr. Ibn Kathir says in Bidayah Wa Nihaya, "Allah Ta'ala mentions him in the Qur'an but not by name, in the story of Khidr, when he says, 'And when Musa said to his young companion,' and 'and when they reached it, he said to his young companion,', the prophet (SAW) clarified that the young companion is Yusha Bin Noon in a Saheeh Hadeeth," (Bidaya Wa Nihaya, volume 1, p: 378)
This can be found in the following authentic Hadeeth:
"Prophet Muhammad (SAW) said, 'Once Moses stood up and addressed Bani Israel. He was asked who was the most learned man amongst the people. He said, 'I.' Allah admonished him as he did not attribute absolute knowledge to Him (Allah). So, Allah said to him, 'Yes, at the junction of the two seas there is a Slave of Mine who is more learned than you.' Moses said, 'O my Lord! How can I meet him?' Allah said, 'Take a fish and put it in a large basket and you will find him at the place where you will lose the fish.' Moses took a fish and put it in a basket and proceeded along with his (servant) boy, Yusha 'bin Nun, till they reached the rock where they laid their heads (i.e. lay down). Moses slept, and the fish, moving out of the basket, fell into the sea." (Saheeh Bukhari 4: 55:613 )
Yusha Bin Noon (AS) was also Musa's (AS) successor. When Musa (AS) passed away, he became the prophet and it was under his leadership that Banu Israel finally conquered Jerusalem.
The prophet Muhammad (SAW) mentions one miracle of his. Yusha wanted to conquer Jerusalem before sunset so Allah cause the sunset to be delayed until he had conquered Jerusalem. Here follows the complete narration:
It has been narrated by Abu Huraira that the Messenger of Allah (may peace be upon him) said: "One of the Prophets made a holy war. He said to his followers: One who has married a woman and wants to consummate to his marriage but has not yet done so; another who has built a house but has not yet erected its roof; and another who has bought goats and pregnant she-camels and is waiting for their offspring-will not accommpany me. So he marched on and approached a village at or about the time of the Asr prayers. He said to the sun: Thou art subserviant (to Allah) and so am I. O Allah, stop it for me a little. It was stopped for him until Allah granted him victory. The people gathered the spoils of war (at one place). A fire approached the spoils to devour them, but it did not devour them. He (the Holy Prophet) said: Some of you have been guilty of misappropriation. So one man from each tribe should swear fealty to me. The did so (putting their hands into his). The hand of one man stuck to his hand and the Prophet (may peace be upon him) said: Your tribe is guilty of misappropriation. Let all the members of your tribe swear fealty to me one by one. They did so, when the hands of two or three persons got stuck with his hand. He said: You have misappropriated. So they took out gold equal in volume to the head of a cow. They-placed it among the spoils on the earth. Then the fire approached the spoils and devoured them. The spoils of war were not made lawful for any people before us, This is because Allah saw our weakness and humility and made them lawful for us." (Saheeh Muslim 19:4327)
We learn that the prophet mentioned in the above Hadith is Yusha from the following narration mentioned by Imam Ahmad Ibn Hambal with an authentic chain of narrator: The prophet (SAW) said, "The sun has never stopped for any man besides Yusha when he wanted to conquer Baitul Muqaddis," (Bidaya Wa Nihaya, volume: 1, p: 382)
According to Islam Online, the name means "God is my salvation"
You see, even naming a child can be educating. :)