Why did Jesus come from the tribe of Judah?

Paul Miller's book on prayer, A Praying Life, brought up the question, why was Jesus a descendant of the tribe of Judah? Why did God choose to use this tribe to fulfill his purpose?

For some background we can look at when Jacob blessed his 12 sons near the end of Genesis he pronounced this blessing over Judah,

 

"The scepter shall not depart from Judah,
    nor the ruler's staff from between his feet,
until tribute comes to him;
    and to him shall be the obedience of the peoples." Genesis 49:10.

So we have a prophetic statement that Judah's tribe will the ruling tribe as the "the scepter will not depart from Judah."

I've wondered if there is any character trait in Judah that foreshadows who Jesus is. The reason I believe this is a possibility is because of at least two verses. One of them is John 5:39 where Jesus is talking about how the Old Testament scriptures offer eternal life because they testify about him, "You search the Scriptures because you think that in them you have eternal life; and it is they that bear witness about me." Jesus makes the same point again after his resurrection, "And beginning with Moses and all the Prophets, he interpreted to them in all the Scriptures the things concerning himself." Luke 24:27 So with these two verses we have a starting place to go back and visit the Old Testament to see what matters deal with Jesus.

After reading Paul Miller's book, I went back to Genesis 44 as I considered his question of "why did the Messiah come from the tribe of Judah?" The setting of Genesis 44 is that Joseph is in power ruling as second in command of Egypt. His brother's have come to buy food there and they haven't yet recognized him as their long lost brother. So Joseph takes this as an opportunity to test them. 

Joseph sets it up so it appears that his brother's have stolen from him, thus earning his punishment. Joseph offers to make only the individual brother "responsible" for stealing to become his personal servant by saying, "...Only the man in whose hand the cup was found shall be my servant. But as for you, go up in peace to your father.” Gen 44:17. 

But Judah does something remarkable. Judah offers to become a servant in place of his younger brother who has had the evidence planted on him. Judah said, "Now therefore, please let your servant remain instead of the boy as a servant to my lord, and let the boy go back with his brothers" Genesis 44:33 (emphasis mine.) Judah has offered to take on a punishment of servant-hood so that his brother can go free and his father wouldn't weep over losing his remaining favorite son.

We see that it is Judah interceding for his younger brother that most foreshadows and teaches us something about Jesus. Jesus takes our place on the cross. We deserve punishment. The criminal charges against us aren't faked. We reject God and his ways. We fail to love our fellow brothers and sisters. We are guilty. But Jesus goes to the cross instead of us. 

So that's one insight I've gained from my reading. Intercession is one motivating factors for why Judah and his tribe would be the particular family that would one day give birth to the Messiah. Intercession is about intervening, and stepping in on behalf of another.

If anyone knows of clearer verses that more directly explain the motivations of God for using the tribe of Judah in this way, I'd love to hear about it.