Most cultures deeply value family in some way. The value of family isn't actually cultural; it's a human value.
That being said, when my family and I moved to the Middle East, "family ties" took on a whole new meaning for me. I could not believe how many students at my school knew how their aunt's grandfather's brother was so-and-so's grandmother's nephew's in-law. Every weekend, one of my best friends would have family "reunions" with her extended family—both sides. As an Asian American with most of my relatives living in different parts of America and Korea, I was lucky if we got in a family reunion with at least one side every couple years.
But even though you and I may not express our family loyalties as intensely or extensively as people do in Middle Eastern cultures, there is no doubt we do love our families. They are the ones who dealt with us during our angsty teen years, who comfort us during heartbreak and disappointments, and who can be counted on to celebrate our most minuscule successes the most obnoxiously. They've made us who we are in the same way we've shaped who they are. Family isn't something you easily give up on, even when it gets hard. In some cases, the same can be said for very close friends.
So, if family is this precious gift that holds immense value to us, what do we do with Luke 14:25–26? Jesus said to a crowd, "If you want to be my disciple, you must hate everyone else by comparison—your father and mother, wife and children, brothers and sisters—yes, even your own life. Otherwise, you cannot be my disciple."
Since this is Jesus, we can't explain this away with an "overexertion-in-ministry" diagnosis, or prescribe him some time in his solitary prayer place. How do we address this? The first thing we need to remember is that family is important to Jesus. He advocated obedience to God in honoring one's parents, and even appointed one of his disciples to be his mother's caregiver as he died. So why is he calling us to hate them?
Jesus is challenging our traditional ideas of family. He is calling us to love him more than our blood families. As contradictory as it sounds, we love others best when we love God most. When we love God most, we seek to please him by obeying his commands. We can see that play out in our own family relationships.