This question seems like it would have a straightforward answer: Jesus' last name is Christ. While today we make this association, if we traveled back to Jesus' time, no one would associate Christ as being a last name in the sense that we understand it. Christ isn't a last name, but a Greek form of the Hebrew title Messiah or anointed one. While some would come to believe in Jesus as the Messiah, or Christ, people wouldn't popularly refer to Him in this way.
One way people have referred to Him reflects a custom that we maintain today. In St. John's Gospel during the Bread of Life discourse as people grumble against Jesus teaching, they ask, "Is this not Jesus, the son of Joseph" (John 6:42)? Just as many choose to give a child the last name of the father, so people might have referred to Jesus as being the son of the one whom they believed to be His father: St. Joseph.
Another way people have referred to Jesus is by the town He grew up. For example in Mark 10:47, St. Mark refers to Jesus as Jesus of Nazareth. Some may have known Jesus by calling Him this name.
The purpose of having last names is to identify what family we come from. There may be many people named John but many fewer named John Mayo, for instance. My full name helps to let people know who I am and distinguish me from the other Johns they may know.
Jesus, too, is a unique individual. Calling Him Jesus Christ helps to distinguish Him and illustrate our belief in Him. As we know though, a person cannot be summed up by their name. This is particularly true for the Son of God. To help us understand better who Jesus is, we have the prayer form of a litany.
At first glance, a litany is just a long list of titles given to Our Lord, Our Lady, or to a saint. But as we prayerfully read over a litany, we recognize that these names and titles illustrate or explain part of who Our Lord, Our Lady, or a saint is. Thus, we gain a greater knowledge and understanding of them personally. Further, perhaps one or two of these lines stand out to us. This may be because that particular aspect is more meaningful to us for some reason or that God wants us to concentrate on that aspect in our spiritual life. Having a litany for us to pray is not an out of fashion, but a real way for us to get closer to God, to Mary, or to a saint and a way that God can speak to us as well.
Father Mayo is pastor of Holy Rosary Parish in Warrenton.