Many people have thought about why Jesus selected the twelve Apostles that he did, and the disciples that became his core group who accompanied him. People wonder about the very ordinariness of the Apostles (let us leave the question of Judas aside for now).
To better understand you need to think about the context. Jesus came as the Messiah to bring the New Covenant to the Jewish people who already had the well established Old Covenant. Secondarily the Apostles were charged with preaching to and converting the gentiles. The time of Jesus' active ministry was only around two years. It was important that everything he said and did was presented in a way that would live on after him. It was essential that his time not be cut short either, as short as it already was. So it was the very ordinariness and "salt of the earth," what we would call "blue collar" traits of the pious men of Jewish faith that recommended them to Jesus. Jesus did not need intellectual gurus to argue with him in order to inflate their own roles and consume his time and authority. He did not need hot heads to provoke armed conflict before his time was due (and even several of these level headed Apostles had had enough when a town was rude and impertinent to Jesus and they questioned why he did not call fire down upon them).
A Covenant is like a marriage. There is a getting to know period so that trust and comprehension can be established. Moses brokered the Old Covenant between God and the Israelites. (In other words, Moses translated God's instructions to the Jewish people, and asked questions of God when circumstances arose.) Jesus TAUGHT the New Covenant through his words and examples. This is what Pope Benedict XVI meant when he wrote in "Jesus of Nazareth" that Jesus brought God to mankind. The intentions of God shone through the words, deeds, and miracles of Jesus. It was not a period of codifying an agreement and rules for life in the way that Moses brokered with God the Old Covenant for the Israelites.
So Jesus needed people who were pious, but not extreme, inquisitive but not argumentative, thoughtful but practical, were "of the people" yet "lived for God." And he needed people who were solid, sane, men with whom he could be friends (as Jesus told them, because he kept no secrets from them, they could see they were his friends and not his servants). He did not not need people who would constantly test him and erode the time they had together by trying to change his will (and this was the temptation of Judas, who did have a challenging and testing streak that would be his downfall). Judas was the only one of the Twelve who had the obviously mistaken inkling that he might "know" more about what God would "want" than Jesus (his comment about how the ointment given to Jesus instead should have been sold for the poor. Have you ever thought of the arrogance of that question, as if Jesus is not fully in communion with God about what material and reverential attentions he was entitled to in life?)
Also, everything that Jesus said and did pointed back to God. Jesus never sought nor would have permitted personal worship of him by his companions, since this would have detracted from his intention, which was to point all piety and worship back to God and God's Kingdom to come. Jesus needed people who would be able to sit next to him and relate to his humanity, not his divinity. Jesus wanted people to detect the light that shone through him as being from God and at God's will and pleasure, not a biological or hereditary imperative. Jesus was very successful at choosing people who kept a level head around him. Humorously, these very Apostles are somewhat nonplussed, even in their amazement, when the resurrected Jesus conversed with them and even broke bread with them. While they were amazed they were not trapped in a cult of personality whereby they were no longer able to worship the one true God of "afar" because of Jesus in his glorified body there with them. These were amazingly balanced and sensible men, aided by the fact that some knew him throughout his life, but also because despite their questions and sometimes puzzlement "they got it." They understood that God shone through Jesus, but that they were not to treat the body of Jesus as though God was walking among them and absent from the rest of the world. They never lost sight of the reality of the all knowing and seeing God who was to be the exclusive focus of their worship, even as they loved and obeyed Jesus, who could call God his Father by being the most beloved of God, and by being born of Mary overshadowed by the Holy Spirit. So the Apostles were able to treat Jesus with the dignity and reverence that he was entitled to, but they never lost sight of the true one God who shone through Jesus and who was to be their lifelong focus for salvation. For example, there was an attempt to relate Jesus to God as King when the mother of two of the apostles asked that Jesus would grant her sons the right to sit at his left and right hand when he comes into his Kingdom. Jesus explains that they can walk his path (and hints this would not be of earthly power, but of persecution) but that God rules heaven and seats there who he will.
Jesus also thought ahead to the time when he would be gone from earth and the Apostles and disciples would have to grow this New Covenant that in a sense Jesus had "given birth to," but had not yet taken its first steps. He wanted people who were of solid, stable, and sensible character who could be persistent and strong in the face of persecution, yet could reach out to the people when evangelizing. Jesus knew that he would request that God send the Holy Spirit to the Apostles, the disciples, and Mary, and so this balanced and strong character was needed to be a worthy receptacle of the Holy Spirit. The Holy Spirit was therefore able to inspire, guide and instruct the Apostles, disciples and Mary without risking an over reaction based on ego or ambition from those who received the Spirit.