This concept of making someone more broadly known is closely related to the Greek meaning of "a name," onoma. This concept is related to today's use of the word, but it is also something more.
In any language, "a name" is the symbol for a mental image of a person. The name just points at the mental image. This image is incomplete and imperfect. No one--other than God--knows another person completely. Different people have different images of the same person. A person's name is a shared common denominator. We use it to "point" at the mental image, even when the images are completely different.
In a Name
There are thirteen times that Jesus uses the phrase "in my name" in the form of "ἐν τῷ ὀνόματί μου", that is, "in that name of mine" where the "name" is a dative (the case is important). This refers to the others acting in his name. The en here has many of the same meanings as is below with the accusative, but Jesus separates them by using dative with this preposition and accusative with the eis below.
However, in Matthew 18:20 ("For where two or three gather in my name"), Jesus uses a different phrase, " εἰς τὸ ἐμὸν ὄνομα" that its better translated as "for this my name", which uses a different pronoun and the accusative, which gives it a different meaning. He also uses the "εἰς ὄνομα" version in Matthew 10:41 and Matthew 10:42. So its sense is "for the reason" of that name or in order to honor a name.
Name and Reputation
In ancient Greek, a person's "name" was more important than the way we think of names today. A person's name was tied to his or her reputation, how that person was valued by other people. This difference was largely cultural.
Since we live in a more anonymous society today, names become like identification numbers. People can know our names even when their only mental image of us is the name on a piece of paper. People can know our names without knowing anything about us or our reputations.
However, in smaller societies, where individuals are known to their community, their "name" becomes their public reputation, their public identity. In Christ's era, even if you didn't know an individual personally and that person was in your community, you knew his or her "name." A person's reputation in the community was more important than anything else. It determined how people treated them, especially those who would do business with them and how much they could earn. Having a "bad name" meant being an outcast.
Even though we are anonymous in many public aspects of our lives today, we still "make a name" for ourselves where we work, within our families, and among our acquaintances. However, this reputation is more limited than the "name" of Christ's time. There are many aspects of our lives that most people won't know. People at work won't know our reputation within our families. Our families won't know our standing amount our fellow workers. Some "important" pieces of information, such as our bank balance, no one may know.
In Jesus's time, however, much more was captured in a person's name, that is, their reputation. People would know about your family even if they didn't know your family personally. In Christ's era, this was the concept of belonging to a "house", which was so important that it would take another article to explain it. People would know where you worked and something about those with whom you worked and that would have a lot to do with your "house." There were no bank accounts, but people would know what you owned and who you owed.
Name and Authority
The Greek word "name" also means a person's "fame" and their authority. We use the word "fame" to describe someone who is known broadly. In ancient times, people would simply say that their name was known, for better or worse, in different areas. When it came to authority, a ruler's name was his authority. He extended his authority by giving others the ability to "act in his name." A ruler's power extended to the extent those acting in his name was respected. Acting in someone's name means using the authority or power of that person. This very much applies to the idea of God's name and Christ representing that name.