(with examples from Harry Potter, The Hunger Games, Lord of the Rings, Trigun, Divergent, How to Train Your Dragon, and more.)Hmmm, names can be tricky. Setting and origin is an important factor. If I lived in China, my name might be "Li Ming Fan." Names like "Alohilani Smith" suggest a mix of ancestral heritage or perhaps the character's mother is Hawaiian and the father American. I have one friend who has a Native American name and she's super pale, yet she has Native American blood, so you could even play around with things like that to make your character a little unique.
Names can suggest culture and ethnicity. Never in the Harry Potter books does Rowling describe the Parvatis' skin color. She just gave them Indian names, Padma and Patil. In The Maze Runner series, we meet Fernando. Dashner doesn't need to tell us Fernando's ethnicity.
Names can also establish time period. I had grandmother named "Merle." No one names their kids that these days, but if you wanted to write a story that took place about 80 years ago or you're writing about a grandma, it'd be fine, just like "Gertrude." Some of those names are coming back in fashion. "Emma" used to be an old name, but in the last decade or so it's become one of the most popular baby names. For my generation, everyone was named "Jessica" or "Megan," so those name would be great for someone my age.
A lot of people are named after public figures or ancestors, so you can think about that. I've known people named after movie stars or prophets in scriptures. It can tell a lot about the character's parents or the character's family. I've heard of one family where the youngest child got to name the next baby--one can only imagine how that turned out.
Maybe your character's parents are movie buffs, so they name their kid "Harrison," after Harrison Ford, because they want him to be an actor. Does the boy want to be an actor? How does he feel about being in a family pushing him into that? That's something you could explore. What about a character named after a prophet? Does he feel inadequate because he's not saintly? What if he's atheist? Would he try to go by a different name? Or wear it more proudly out of spite or snark?
You can pick names based on their meanings. J.K. Rowling does this all the time. "Dumbledore" is Latin for "Bumblebee," and she picked it because she pictured Dumbledore to be the sort of person to hum when he walked about the castle. "Lupin" relates to "wolf," and he's a werewolf. "Fawkes" relates to Guy Fawkes, which is fitting for a phoenix who catches flame. Many of the pure bloods are named after stars, like "Bellatrix" and "Draco," and Sirius is named after the Dog Star, and he can turn into a dog. Suzanne Collins named her character "Katniss," which is an old word for arrow or arrowhead--something like that.
You have to be careful not to get carried away though. Characters named "Faith," for example, who are defined by their faithful characteristics in the story can be pretty heavy-handed. ("Peeta" can be a kind of heavy-handed, since he's a baker, but at least Suzanne didn't spell it "Pita.") BUT, you can look for a name that means "faith" or "bread" in another language, to give it some distance.
In a show I like called Trigun, the villain's name is "Knives." That's his real name. It sounds pretty heavy-handed, until you think about it. Trigun came out of Japan, so they picked a telling English word for their villain's name. It doesn't sound heavy-handed to them, just like "Voldemort" doesn't sound heavy-handed to us, but it's French for something like "He who flees death." And when I figured that out, I was blown away because his desire to conquer death is his biggest aspiration and so "Voldemort" is so fitting.
With all the thought J.K. Rowling put into her characters' names, I like that she picked a fairly common, meaningless, boring name for her protagonist: Harry Potter. She took a common name, that had no special meaning attached to it, and made it the most famous name of her wizarding world (and it became one of the most famous names in our world too.) Harry makes his own name. You can think about doing the same thing, making a meh name into one that means something.
Likewise, I like when characters are supposed to be invisible, so they take on common names. There is a movie called Mr. and Mrs. Smith, about two undercover agents who are married. Of course their last name is "Smith" if they are undercover--that's one of, if not the most, common last name in the U.S. Another example is "John Smith," the protagonist of I am Number Four, who is an alien-in-hiding in the U.S. Can you get more common or boring than that?
You can pick weird names, like "Hiccup" from How to Train Your Dragon. It can add character, but you'll want to address at least once, somewhere that the characters know it's a weird name, otherwise it will test the audience's suspension of disbelief. And, the audience will probably want to know how that character got that name. You'll probably want to mention (or show) whether or not your character likes her weird name.
Consider how the name sounds phonetically. Names with lots of "s" sounds can sound softer or sneaky or sensual. Names with b's and g's and other similar consonants can sound more burly or tough, at least to me. Some names sound more geeky to people and others sound cool. Tolkien paid a lot of attention to how words sound in his Middle-earth world. Listen to the soft and elegant names of the elves: Legolas, Galadriel, Thranduil. Compare them to Hobbit names: Frodo, Bilbo, Pippin, Sam, Merry. While you can study the phonetic sounds and their effects on us as human beings in an English college class, it's important to note the feelings some names evoke are just based on our personal experiences and preferences.
Then, of course, there are nicknames. Does your character have one? Usually a nickname speaks to a particular character trait. In Divergent, Tobias has the nickname "Four" because he has only four fears.
Whew, with all that said, many authors just pick names without much thought.
For me, sometimes I need to try out the name for a few chapters to see if I warm-up to it or not. I had one college friend who threw in the name "Jose" or something of the sort for his Mexican character. He meant it as a place-holder until he found a better name, but by the time he'd finished the story, he'd warmed up to it so much he didn't want to change it. So sometimes it helps just to try one out.