What does Luke 7:8 mean?
A centurion has expressed his belief that Jesus can heal his servant without being physically nearby. To explain, he mentions examples of his own authority. In fact, the entire incident shows the centurion's authority: he told the city elders to go to Jesus and ask Him to heal his servant, and they did (Luke 7:3). When Jesus approached the house, the centurion sent friends to speak with Jesus, and they followed his direction (Luke 7:6).The centurion assumes Jesus works under the same pattern. Jesus represents God; what God tells Him to do, He does. And Jesus apparently has authority over illness, as He has healed many people. If the centurion's supervisors don't have to be present to make sure he does his job, and if he doesn't have to be present to make sure his soldiers and slaves do their jobs, and if God doesn't have to be present to make sure Jesus does His job, there's no reason to think Jesus would have to be present to make sure His will is carried out.
Still, it's not clear what the centurion understands about Jesus' power. He has a high enough view of Jesus to assume He will not want to be tarnished by the company of a Gentile. He respects the Jewish culture and religion (Luke 7:4–5). There's no indication he understands that Jesus is God and the Jewish Messiah, but he apparently believes that Jesus is empowered by the Jewish God. If so, Jesus is obviously holy, and the Jewish God can do whatever He wants.
The centurion's comment about his servant may tie back into the servant's description of "highly valued" (Luke 7:2). Some modern scholars claim the description is personal and attempt to warp this into evidence the two had a sexual relationship. Such interpretations ignore the scholarly process, which moves from specific words to the passage, then the book, other works by the author, then the entire Bible. Other useful references are writings of the early church and secular contemporaries. One such reference, in the text Shepherd of Hermas, cites a Roman philosopher using "highly valued" to mean a servant who went beyond the bare minimum of service. In short, nothing in the story, text, or related culture implies that this passage has any overtones of same-gender relationships.