Using (or calling on) the name of Jesus was a sign of his early followers. It was both a literal term (in that Christians used the name to perform miracles) and figurative one (as it strongly identifies Jesus' followers).
It's probable that this way of identifying believers comes from the first volume of Luke-Acts:
John answered, “Master, we saw someone casting out demons in your name, and we tried to stop him, because he does not follow with us.” But Jesus said to him, “Do not stop him, for the one who is not against you is for you.”—Luke 9:49-50 (ESV)
(It's not obvious from the way the New Testament is commonly arraigned, but Acts and Luke were written by Luke on commission from Theophilus. Acts 1:1-2 explicitly labels itself as a sequel.)
Luke is repeating an even earlier tradition that's recorded in Mark:
John said to him, “Teacher, we saw someone casting out demons in your name, and we tried to stop him, because he was not following us.” But Jesus said, “Do not stop him, for no one who does a mighty work in my name will be able soon afterward to speak evil of me. For the one who is not against us is for us. For truly, I say to you, whoever gives you a cup of water to drink because you belong to Christ will by no means lose his reward.—Mark 9:38-41 (ESV)
The context of this particular story seems to be someone (who was not part of the disciples' group) using the name "Jesus" to perform miracles. We see an examples of this practice a number of times in Acts, including:
But Peter said, “I have no silver and gold, but what I do have I give to you. In the name of Jesus Christ of Nazareth, rise up and walk!”—Acts 3:6 (ESV)
Peter emphasizes in Acts 3:16, that the man was healed "by faith in [Jesus'] name". It seems that this very act was what prompted the Jewish leaders to later give Saul the authority to bind believers:
On the next day their rulers and elders and scribes gathered together in Jerusalem, with Annas the high priest and Caiaphas and John and Alexander, and all who were of the high-priestly family. And when they had set them in the midst, they inquired, “By what power or by what name did you do this?” Then Peter, filled with the Holy Spirit, said to them, “Rulers of the people and elders, if we are being examined today concerning a good deed done to a crippled man, by what means this man has been healed, let it be known to all of you and to all the people of Israel that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead—by him this man is standing before you well. This Jesus is the stone that was rejected by you, the builders, which has become the cornerstone. And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.”—Acts 4:5-12 (ESV)
While the actual case was about a man healed using the words "In the name of Jesus Christ of Nazareth...", the concern of the Jewish leaders was whether the healing was powered by God or by some demonic force. The conclusion of the trial was that the leaders "called them and charged them not to speak or teach at all in the name of Jesus." (Acts 4:18 ESV) So this seems to be more figurative: using the name of Jesus as a sign of authority.
The passage you mainly ask about is instructive in the way that it uses the word name:
Now there was a disciple at Damascus named Ananias. The Lord said to him in a vision, “Ananias.” And he said, “Here I am, Lord.” And the Lord said to him, “Rise and go to the street called Straight, and at the house of Judas look for a man of Tarsus named Saul, for behold, he is praying, and he has seen in a vision a man named Ananias come in and lay his hands on him so that he might regain his sight.” But Ananias answered, “Lord, I have heard from many about this man, how much evil he has done to your saints at Jerusalem. And here he has authority from the chief priests to bind all who call on your name.” But the Lord said to him, “Go, for he is a chosen instrument of mine to carry my name before the Gentiles and kings and the children of Israel. For I will show him how much he must suffer for the sake of my name.”—Acts 9:10-16 (ESV)
It seems that the primary sense of the word, as used here, is to identify someone or something. Therefore, the primary sense of "calling on the name of the Lord" is to identify someone as part of the group who follows Jesus.