I have been playing tabletop rpgs for quite a while now, mostly Dungeons and Dragons, and I've tried every version of D&D so far, and a few games inspired by it, and ever since the beginning, rolling dice has been the default method for building a character... but this is a bad idea.
Rolling for Attributes
Typically, when creating a new character for a campaign, the first thing you'll do is roll a bunch of dice to determine your attributes. This can be fun and exciting, but this is also very unbalanced and terrible. According to the rules of D&D 3.5, you roll 4d6, ignore the low die, and keep the remaining total. This becomes one of your attribute scores. You do this five more times and assign all six scores to the attributes you want. If none of your scores is higher than 13, you may scrap it all and start again. If the total attribute modifiers add up to 0, you may scrap it all and start again.
These rules do give you at least some form of safety net to keep your character from being totally useless, but if you end up with scores of 14, 13, 11, 10, 10, 7, then you do have an attribute that's higher than 13, and your modifiers will add up to a total of +1, and when you find out one of the other players ended up with 18, 16, 14, 14, 12, 10, you're still going to feel pretty useless.
This is the problem with rolling for attributes. There is no balance. Tabletop rpgs are a group game. Whenever one player gets more than another player, the player with less is going to feel like they aren't contributing as much to the group as the player with more.
Rolling for Hit Points
Every version of the d20 system I have played so far has a "hit die" for each class, that tells you what die you roll to add to your hit points at each level. In most of these, fighter or soldier classes tend to have a d10 hit die, while skillful, rogue-like classes tend to have a d6 hit die, and then other classes will have a d8 or even as low as a d4. These dice tend to be distributed among the classes based on how often or how likely the class should expose themselves to combat. For example, if you are playing a d10 class, your character is probably suited for getting into the thick of things, while a d6 class is better suited for lurking at the edge of a fight and looking for opportunities to attack.
Games like this use guidelines to help the GM determine if an encounter is balanced for the party. Typically, these guidelines assume an average party. What happens if the party's front-line fighter rolls a 1 for his hit points? What happens if he does this for two levels in a row? Now the character who is supposed to handle most of the fighting no longer has enough staying power to fight opponents of his level.
Randomness is Bad
In any game system that uses random elements, such as dice, these elements work against the players. Player characters go through several game sessions, possibly an entire year or longer of making die rolls or whatever other random mechanic your game uses. Your player characters have hundreds of opportunities for failure, but the average NPC (non-player character) is only around for one encounter. If the NPC fails, who cares? You'll just use another NPC in the next encounter.
Any random element is a potential for player failure. Randomness should only occur DURING gameplay, when determining if players succeed or fail. Players should not have to deal with any amount of randomness while building their character. Think about this, a failed attack roll or skill check only hinders a player for a moment, but a low hit point or attribute roll hinders the player for the entire campaign.
A Better Way
I have been phasing out character build randomness from my games. I've been using the point system for attribute scores, and have been giving players the option of taking an average result for their hit points instead of rolling. The average hit point rule suggested in the Unearthed Arcana says that players should round down on every odd level and round up on every even level, so a d6 hit die would give 3 hit points or 4 hit points depending on what level you attain. I decided to be a little nicer, and also not make my players keep up with whether they're rounding up or down, and just let them round up at every level, which results in higher than average hit points, but I don't care if my players have a few more hit points, it doesn't break the game.
To calculate the average of any given die roll, take the lowest possible roll, add the highest possible, and divide by 2. For a single die you get the following: d4 (2.5), d6 (3.5), d8 (4.5), d10 (5.5) and d12 (6.5).
Now, of course, if everyone has the same points to choose their attributes and has average hit points, you might think that everyone will feel pretty much the same. While it's true that using this system doesn't allow for as much variety, in a game, it is more important for everyone to have the same potential than for a player to have the feeling of getting a lucky roll and starting with higher attributes. When I allowed players to roll for attributes, I would often try to counter a player's bad luck by allowing everyone to roll more than one set of attributes and choosing what they wanted. It was my intention to allow players to have the fun of rolling without condemning a player to their bad luck, but in practice, this has more often resulted in players with good luck getting even more outrageous results than keeping the party close to even. So, it's really just better to cut out rolling altogether if you want your players to be able to build characters of similar ability.
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