Your name is an important social construct throughout the world. For Westerners and Easterners alike, it can symbolize important relations, monetary status, reputation... Entire careers have been built on names (Nick Carter begat Aaron, Jonathan Swift begat Taylor), favors have been traded, etc. But, outside of a social context, Western names don't really mean anything. Yes, the etymology of Andrew means something along the lines of "Strong," but the name as it stands in modern times is really meaningless. I don't meet people named Rex and think, "Yes, for he truly bears the noble stature and hard nose of a great king." That I don't meet people named Rex at all is beside the point.
In the East, names still mean something. It is unavoidable in Chinese, as the characters are identical, and so a link of meaning occurs. Parents will often name their children for an aspiration or a hope. I have a friend at my school named 豆豆, which means "Bean sprout." I've yet to crack what the aspiration there was.
Many Chinese people fail to grasp that this does not work both ways. When starting a class of new learners, many of them will not have 英语名字 (English names), and so we bequeath these upon them. The parents will often ask after the class for the meaning of the moniker.
"His name is Rick?"
"What does it mean?"
"It means Rick."
"But what does it mean?"
This can go on for ages. As teachers, we strive to create a controlled, respectful learning environment for all of our students, but I think the cracks show through when we name our students. In my first few weeks of teaching, I attempted to name a student "Ishmael" and another "Cain." Both ultimately rejected these names for "Stephen" and "John," respectively. I understand why Ishmael jumped ship, but, for God's sake, I would love to be named "Cain". I briefly flirted with naming another kid in the same class "Abel," but I didn't want to be asking for trouble. You can only do so much. One teacher recently gave two students the names "Sonny" and "Cher." They will, I assume, be partnered up in all pairs activities for the foreseeable future. There was a student recently who named himself "Airplane". The TA for the class tried to tell him that that's not a name, but the teacher said, "No, if he wants to be called 'Airplane,' let him. This is how we learn."
Funny in a less troubling way are the names students come in with. As all nouns are considered fair game in Chinese, you do end up with some classics: I have no fewer than three "Apples" in my two youngest classes. There's a "Linvida," which I still insist is not a real name. There are quite a few "Potato"s, a "Power," my roommate is teaching a "Dinosaur," and there's even a kid named "Michael Jackson," which really makes you wonder. There's a "Star" who consistently introduces herself as "Superstar." I refuse to call her this.
My own experience with picking a Chinese name hasn't been what I would call "Quite a process," but it's been a source of bemusement on the occasions when it's come up. My Chinese tutor suggested 安林杰, which is derived from my first and last names to give us "An Linjie," but that has met with general scorn from every other person in China. Taken literally, it means "Peaceful, talented grotto". But more recently, another option has presented itself in the form of 安猪, "An Ju". This is simply another phonetic transposition of my name. There are many, many possible ways of rendering my name in Chinese similar to this. But 安猪 is special. It's Chinese for "Peaceful Pig." Winner.