It may seem a little weird to devote one chapter of the Mary Sue series to names, but I think I can tell you a few things you've never thought of before.
First of all, with a Mary Sue, the name tends to be exotic, like Angel or Raven, or has an odd spelling like Myshel, usually in a world were Mary, John, Sue, and Amanda are common names. An exotic name is meant to convey that the character is special or worth noticing.
There is nothing wrong with Angel, Raven or Myshel. In the proper universe, of course.
I keep coming back to the proper universe, and I will explain more on that as it's own thing later on.
The thing about names is that there are people who dislike their name for one reason or another. I know I prefer not to use my first name when I can. When we name our characters, we give them the names we wish our parents had given us. In this, our characters are our children. The only difference being that most characters will be indifferent to love their names and our children would probably disown us if we named them Princess Raven Margretta Kristina.
Another thing is spelling of a name, while 'Myshel' might make since to me as being pronounced like 'Michelle', someone else might read it as 'My-Shell'. An unusual spelling of a name tends to lean more towards being a personal headache than being a cool idea. Especially in a world with Mary, John, Sue, and Amanda. Of course there's always Sean versus Shawn. Believe me, that one trips me up ninety-nine times out of a hundred.
The trick to a name is to make sure that it fits in with the culture and race of the character. If you have a girl from Russia a name like Natalia will play better than a name like Chang Li.
Nicknames don't really have the same constrictions. They're either short versions of the character's name (Tam for Tammy, Mike for Michael) or they play some meaning on the character as a person (Pocket Rocket, he's short and fast). While they can be cultural in nature, they don't necessarily have to be, like a Japanese-American teen being called Deke because he's the fourth born, only son who loved Mighty Ducks as a kid. (Live-action movie, the triple deke was a hockey move, three fakes and the score if I misremember correctly).
To find a good name, there are numerous sources, particularly baby name sites. My personal favorite for name hunting is called behindthename.com which features the history and etymology of first names. It has a sister site called behind the surnames for last name hunts. Also useful if you're being culturally accurate.
In short, you can call your character any name you want, but the less exotic to the character's culture, the better. A name will work, in the world it's intended to work in. An outlandish name in a mundane world does not work.
Up next, the Mary Sue Universe