I don't think "family" meant Family as we read it;
This seems worth reading:
http://www.the-churchofchrist.com/passage_study/would_you_explain_the_meaning_of_matthew_103436_as_quoted_from_micah_76.htm
Now, how does this apply to
Matthew 10:34-36? If you go back all the way to the beginning of the chapter you will notice that this chapter concerns the sending of the disciples on the limited commission.
The entire chapter is devoted to instruction on what the disciples should expect when they go out into the world and preach the gospel to the "lost sheep of the house of Israel" (verse 6). There are two types of instruction given to the disciples in this chapter. First there is positive instruction as to what they are to do on the limited commission. This comprises verses 5-15.
Then there are various and sundry warnings given to the disciples about what they could expect from the unbelievers. This comprises verses 16-42. Chapter 11 verse 1 clearly indicates that this entire section of scripture was given as instruction for this limited commission because it gives this conclusion to the commission in these words: "When Jesus had made an end of commanding his twelve disciples."
The section of scripture with which we are concerned falls within the sundry warnings that Jesus is giving to his disciples regarding the lost sheep of the house of Israel. Jesus words to them are basically this. Don't expect that you will receive a warm welcome out of the unfaithful. When you preach the gospel, people are going to become your enemies. In this sense, Jesus' came to bring a sword. The warning of Micah describing the unfaithful holds true here. Even among families, there will be division. This is the typical response of the unfaithful. So Micah's description of the unfaithful apply to the situation where disciples would be going into the very houses of the unfaithful to preach the gospel, and they were to be aware as to how the unfaithful would respond to this preaching. So in essence, that is the reason Jesus quotes from Micah.
One further question presents itself. What applicability does this passage have, if any, today?
While the passage specifically addressed what the twelve disciples were to do on the limited commission, we can learn from both Micah and from this passage what the attitude of the unfaithful will be. When it comes to preaching the gospel, the unfaithful will not want anything to do with it. They will oppose it. They will reject it. They will even sever family ties over it if they disapprove of it. It is our responsibility, however, to preach the gospel regardless of what kind of problems that may cause. So this passage serves as a warning to us as well regarding what kind of attitudes we should expect from some people in regard to the preaching and teaching of the truth. We should also note that Jesus mentions that some will respond in a positive way to the gospel. Notice verses 40-42, "He that receiveth you receiveth me, and he that receiveth me receiveth him that sent me. He that receiveth a prophet in the name of a prophet shall receive a prophets reward; and he that receiveth a righteous man in the name of a righteous man shall receive a righteous mans reward. And whosoever shall give to drink unto one of these little ones a cup of cold water only in the name of a disciple, verily I say unto you, he shall in no wise lose his reward."
