The Greek word translated as "glorify" and "glory" are from the same Greek root, just like our words "glorify" and "glory" are from the same English root. There translated as "glory," however, is misleading. The English words were chosen more to praise Jesus than to translate his words. At best, the "glory" aspect of these words in tangential.
A more accurate translation of glorify and glory in today's English would be "recognize" and "recognition." However, what is recognized is a person's "reputation." "Recognizing" someone also means spreading their reputation, as we might say, "publicizing" or "promoting." These words provide cover both the primary and secondary meaning of these Greek words as Jesus used them.
The Greek word translated as "glorify" is doxazo, which primarily means holding something or someone in your mind or imagination. Its secondary meaning is to "extol" or "magnify," which is where the meaning of "glorify" as in "praise" arises. In the Latin Vulgate the Greek word was translated as clarifico, which means "to make illustrious or famous." Interestingly, this Latin word is also the source of the English word, "to clarify," which goes back to the primary meaning of the word, having a clear image in your mind.
The noun form is "glory" in English. This is from doxa, which means "expectation" and "opinion." It came to mean "reputation," especially "good repute", "honor", "glory" and rarely "ill repute." It came to mean "glory" and "magnificence" in external appearance through Christian writing after the Gospels were written. Christ did not seem to mean it that way, at least not exactly.
Generally, in the alternative translations of Christ's words on this site, the most common word used to translate doxazo is "to recognize." This idea also works for the noun form as "recognition." This idea of "recognition" captures the mental imaging part of the concept and the idea of giving honor. By giving people recognition, we make them better known and more broadly respected. People who are more broadly recognized are more famous. By giving people recognition, we make them better known and more broadly respected. This is the idea of being a "recognized authority."
To maintain the connection between the noun and the verb, the noun should be translated as "recognition" when possible. While "reputation" is more accurate, "recognition" also has a sense of praise in English. Since Jesus often uses this word only in the sense of a shining reputation, "eminence" sometimes fits better with the context.
Since the verb form also has the sense of spreading someone's good reputation, "publicized," "promoted," and "proclaimed" also work. Today's processes of promotion and advertising are the most common forms of "glorification." Advertising is used to create images in our minds, making people and things more illustrious and famous.
However, in Jesus's era, the process was almost entirely word-of-mouth, much like the communication processes we see on the Internet today: people sharing information with others because it is interesting. The process of making someone or something more broadly known is what Christ describes by doxazo, that is, glorification.
We could use "advertise" or "promote" in the place of "glorify" in today's translations of Christ's word, but many would certainly think of that as disrespectful.
Glorify and Name
A concept closely the idea of "glory" is the word translated as "name" (onoma). Someone's name represented their reputation. It was how they were recognized. A person's name was also a sign of their authority and responsibility so "recognizing a name" is the idea of recognizing someone's authority. When someone "acted on the name of another," they were recognized as representing them. See this article for more about the Greek concept of "name."
Jesus describes his role "to glorify [God's] name" (John 17:6) and to "declare the name" of God (John 17:26). The term translated as "manifest" primarily means "to reveal" and the Greek word translated "to declare" means "to make known." The glorification, that is, promoting God's name, making it famous is the point of in that process (John 17:4). However, it is something that Christ cannot do alone. The Father must make Christ famous so Christ can make the Father famous "glorifying" or "promoting" him on earth. (John 17:1). In today's terms, we might say Christ's job was marketing and promoting God's name. His life was designed to "go viral."
This brings us to the larger question of the "name" of God that Christ is publicizing. What is the mental image that Jesus wants people to have of God? Everything that Jesus teaches is wrapped into this question, but it is best summarized by the term "Father." And it is best express in the Lord's Prayer, "Our Father, in the skies." This is built on the Jewish understanding of the name of God as "I am," that is, the "Being of Existence." However, this is clearly a concept is beyond our understanding. The primary message of Jesus was to make God known in a way we could understand. as a "father," which in Greek means every progenitor in our history, and the highest of fathers, the one in the skies.