Reason for the choice of new 12th Apostle? In Acts 1:20–26, the Apostles are choosing the 12th Apostle who will replace Judas:


  20 Peter continued, “This was written in the book of Psalms, where it says, ‘Let his home become desolate, with no one living in it.’ It also says, ‘Let someone else take his position.’
  
  21 “So now we must choose a replacement for Judas from among the men who were with us the entire time we were traveling with the Lord Jesus—22 from the time he was baptized by John until the day he was taken from us. Whoever is chosen will join us as a witness of Jesus’ resurrection.”
  
  23 So they nominated two men: Joseph called Barsabbas (also known as Justus) and Matthias. 24 Then they all prayed, “O Lord, you know every heart. Show us which of these men you have chosen 25 as an apostle to replace Judas in this ministry, for he has deserted us and gone where he belongs.” 26 Then they cast lots, and Matthias was selected to become an apostle with the other eleven. [NLT]


Does bible say anything about the choice of the new Apostle? What did Matthias have that Judas did not? Does the Bible explain this?
I know the answer will be extremely unsatisfying, but the answer is right there in the question.  Verses 24-26.  They didn't know which to pick, so they prayed and asked God to show them by directing the outcome of casting lots, then they trusted Him to do so.  There's really nothing more said, and really no reason to say more.  

In essence, they threw up their hands and said "We don't know who to choose, God.  We're just going to cast lots and trust you to pick the right man."

Casting lots, if you're not familiar with the term, is similar to drawing straws, or flipping a coin.  It's mentioned elsewhere in Scripture as well.  There's an article here about it:  http://www.biblestudy.org/question/what-is-casting-lots.html