Though the above answers are excellent, I'll be coming at things from a different direction and with a different perspective.
In Middle Eastern culture, and to a certain extent even today in cultures around the world, the responsibility of hosts is to make their guests feel welcome, accepted, and loved. Although Jesus loved Mary, Martha, and Lazarus and had spent a good deal of time with them, He was still a guest in their house.
Which was more important to Jesus: the cleanliness of the house and what was for dinner, or conversation with His friends and His disciples? Obviously, the latter. I suggest that Jesus, instead of having a big meal with His friends and disciples, would rather have fasted (i.e., skipped the meal) and simply conversed with the people who had gathered in the house.
Remember Jesus' words in John 4:
"But [Jesus] said to [His disciples], 'I have food to eat that you do not know about. . . . My food is to do the will of Him who sent Me and to accomplish His work'" (vv.32,34).
Jesus also said,
"'It is written, MAN SHALL NOT LIVE ON BREAD ALONE, BUT ON EVERY WORD THAT PROCEEDS OUT OF THE MOUTH OF GOD'" (Matthew 4:4).
So, which is more important: bread? or words from God's mouth? Since Jesus was the God-Man, Mary's listening to what He had to say was the "better part," as compared to Martha's distracted serving.
When people get distracted, in effect they are forgetting that "the main thing is to keep the main thing the main thing!" Mary got it; Martha, not so much.
The "better part" of hospitality for us Christians today is to make our guests feel welcome, accepted, and loved. In Greek, the New Testament word hospitality means literally "lover of strangers," which concept has its roots in the Tanakh and its teaching on the responsibility of God's people to care for orphans, widows, and aliens (i.e., strangers). That God was concerned about the aliens in Israel's midst suggests to me that God has the heart of an evangelist who wants aliens/strangers to become part of His forever-family.
In conclusion, the main thing for Jesus was to do His Father's will and to finish His work. For Jesus, this involved preaching, teaching, healing, forgiving, discipling, and ultimately giving His life as a ransom for many. Mary realized this and consequently hung on His every word. Martha, though she may too have realized this, she had forgotten it, temporarily, by allowing herself to become distracted, worried, and upset.
When's the last time you skipped a meal to get to know Jesus a little better and to listen to what He has to say to you? His words proceed from the mouth of God.
Food for thought (pun intended).