What does Luke 9:61 mean?
Much of this chapter describes what Jesus expects in His followers. In Luke 9:57–60, Luke presented two flashbacks from earlier in Jesus' ministry. In one, a scribe wanted to follow Jesus, but Jesus warned him he would have to surrender his creature comforts (Luke 9:57–58; Matthew 8:19–20). In the second, Jesus invited one of His disciples to follow Him more fully. When the man asked to bury his father first, Jesus told him to let someone else do it; he needed to teach the world about God's coming kingdom (Luke 9:59–60; Matthew 8:21–22).This last interaction is original to Luke, and we don't know when it happens. Nor do we know what "saying farewell" to his home means. Surely Jesus would let him run into the house, grab his cloak, and kiss his mom goodbye. There must be some tighter emotional connection we're not aware of.
What we do know is that Luke has included several allusions to the Old Testament prophet Elijah in this chapter. When Herod Antipas contemplated who Jesus might be, one of the options was the return of Elijah (Luke 9:8)—a sentiment held by some of the people (Luke 9:19). When Jesus took Peter, James, and John to the top of a mountain to reveal His glory, Moses and Elijah met Him there (Luke 9:30). In fact, Luke mentions or alludes to Elijah more than a dozen times from Luke 4:25 through the end of this chapter.
In this case, Jesus and the man resemble Elijah's call to Elisha. The difference is, Elijah let Elisha say goodbye to his family (1 Kings 19:19–21). Jesus' strong words may represent the fact that His mission is more important and urgent even than Elijah's. Or, as with other responses, it might be a way to deflect someone who simply wants an excuse to say, "no thank you."