This is kind of system specific, but as the game I've played the most for the past several years has been L5r, and it also seems to be the system I've seen people have the most problems with naming their character in, I figured it might be fun to do a post on it. My apologies for the Non-L5R players out there, as not much of this is going to be relevant to non-samurai related games. Though, it could help you with samurai characters in other games, I suppose.
A Name You Choose For Yourself
The first thing you need to understand when you make an L5R character is that the name your character has is one he gave himself. Rokugan does not follow the modern naming conventions where parents name their child at birth. In Rokugan when you go through your gempukku (coming of age ceremony that makes you an adult 'officially') you choose a name for yourself. Because of this, and the ritualized manner in which the naming is done, the character's name should have some significance to them. Perhaps the name of an ancestor they wish to honor, a hero of legend they wish to emulate, or a friend they need to keep a promise to. It could also be something they feel describes them - as a name is supposed to do. Or a reminder to themselves to help keep them on their chosen path. Whatever it is there are two key points here: One, the samurai chooses their own name. Two, that name has meaning to the character and who they are in some way.
Also of note is that, in Japan at least, it was not uncommon for someone to change their names over the course of their lives. Generally, whenever a new facet of hteir life began and sometimes to represent different aspects of themselves. An artist may paint under one name, write under another, run their business under a third, and be known by a fourth by their wife. At the same time, upon mastering a form a student might take on a name that honored their sensei in some way but also branded their own identity and use that name from then on going forward. This is not as common in Rokugan it seems, but it could easily be present if you wanted it to be.
Real Samurai Names
Being that L5R is a game about Samurai, and the key families in the game have names that are often stolen from actual clans, families, and samurai in real world Japan, it shouldn't be that big of a surprise that a wikipedia article on Samurai names, or a list of japanese names, may be a great place to start for a character name. Often these lists will have some inkling of the meaning as well which can help you with finding a name that fits your character well.
Family Names Need Not Apply?
One of the common things I've seen around L5R groups is a player getting crap because their characters personal name is actually a family name in the real world. Now, while your mileage may vary, my opinion is that this is not a valid thought for Rokugan. Why? Well, for two things. One, samurai choose their own names so it is very possible that someone wanted their deeds to honor that family line. Two, when a samurai does something truly great they may be bestowed their own family by either the Emperor or their Daimyo. That family will have their name as it. In other words, in Rokugan, a "family name" becomes a family name when someone who has it as their personal name does something great.
This means that in Rokugan it is very possible that a famous family name like Toyotomi could have started off as Hida Toyotomi or Shinjo Toyotomi before doing something awesome and being made his own family (the Toyotomi family) and perhaps his own clan (the Toyotomi clan).
However, family names that are prominent family names in Rokugan (i.e. Akodo, Shinjo, Doji, etc) should not be taken as personal names unless you truly are arrogant enough to claim you are a god reborn.
Happy With The Sound
The most important thing when considering a name above everything else I've said is that you, the player, should be happy with how it sounds. this is a name you are going to have to say, and hear - in reference to yourself no less - countless times over the course of the game. So make it a name you like the sound of. Something that flows. My head keeps going back to "Akogo Genji" because I really like the sound of that name and flow of it when making a character. Other people have their favored names as well. Ultimately though, Rokugan is not Japan (as much as it is based off of it) so just be happy with your name choice and fit the other stuff in as you can.
Go As In Depth As You Want
The specific part of Japanese history that inspired Rokugan was, iirc, the court heavy years in the 1600s. Rokugan is less based on the actions of Samurai than it is based on the dramatic and romantic tales that Japanese courtiers told during the courts. Because of this there is a very deep level of artistry and scholarly talent that goes into names. Matsu, a common sound in names, means pine. Pine has specific meanings and implications in Japanese folk lore as a wood and the qualities it has for that. So a character named Matsumoto (effectively pure pine, or the basis of pine) would have very specific meanings that a Japanese courtier would understand but your average group of Table Top gamers - even one dominated by Japanese majors, would not likely get.
Because of this, you can feel free to go as in depth or as shallow as you want when naming your character. Obviously you should attune slightly to the game you are in, but dont worry if your name seemingly doesn't make sense and you like it. Odds are with the right amount of time and inclination one of those people who wrote what L5R was based off of could find a way to make it work, and ultimately it doesn't matter because Rokugan isn't Japan and it certainly isn't the real world. So have fun.