As already mentioned by @Affable Geek, the identity of a person is an important thing in biblical times. The importance of identity is also to be able to trace your history back to the promises of God. So for a Jew, it is important to trace your origins back to Abraham to show that you are a child of Abraham and therefore part of the promises that God made to him.
For example, King David, it is important to show that he is a decendant of Abraham because we see God's promises partly fulfilled in Him (Israel has land, has become a great nation (offspring) and bring blessings to other nations).
A promise is also made to David that an even greater King will come from his line
2 Samuel 7:12 (NIV)
12 When your days are over and you rest with your ancestors, I will raise
up your offspring to succeed you, your own flesh and blood, and I will
establish his kingdom. 13 He is the one who will build a house for my Name, and I will establish the throne of his kingdom forever.
We know that Jesus is that king. Part of Jesus' credentials is his anncestory. In fact, this is so important that when Paul reminds Timothy of the Gospel they are preaching he says in 2 Timothy 2-8 (NIV):
Remember Jesus Christ, raised from the dead, descended from David.
This is my gospel
Liniage is also important as it shows the trustworthy nature of God. If Jesus couldn't be traced back to David then either Jesus isn't the King that God promised, or God changed His mind and chose some other person to be the King. If God changes His mind then we cannot trust Him. Yet, the bible shows that God consistently keeps His promises (for example the promises to David) and so we can trust Him.