Letting players express themselves in a game can be a very powerful tool. It can allow players to tailor a game to their whims, whether aesthetic, beliefs or even playstyle.
The key to self-expression is giving players the ability to make meaningful decisions that are decided by their whims. From this, we derive the following
Multiple Acceptable Choices
In order for players to make a decision according to their whims, it is necessary for there to be a decision point with multiple acceptable answers. When you decide what number to put into a square in a Sudoku, there is no scope for self-expression as there is only one correct answer, but when choosing clothes for your avatar, there tend to be many acceptable choices with no gameplay impact and so players can choose the one that they like the most.
Note however that the existence of an optimal choice does not necessarily mean that self-expression is dead. Players will make sub-optimal choices in games if it is in accordance with their desires, but if this conflicts with their desire to win, then they can feel like the game isn't giving them the choices that they want. All of this assumes that they are capable of continuing to play even if they make a sub-optimal choice, unlike in the above Sudoku example.
Amusingly however, it is the whim of some players to win with sub-optimal choices and prove their skill and individuality by doing so. It's just cool to win with something that everyone else thinks is weak.
Another corollary is that difficulty can reduce a player's scope for self-expression as it can reduce the number of options that allow progress. Good balance naturally mitigates this as good balance allows multiple viable options.
A coin flip gives the guessing player multiple acceptable choices, both heads and tails are valid guesses. However, most players don't have a conscious preference for either choice. The decision exists, but the whim does not. On the other hand, choosing between the light side and the dark side in Star Wars: Knights of the Old Republic is something that players have strong feelings for.
When the choice allows players to customize the way they play the game, that can often be depended on for player whims. A game like Magic: the Gathering provides aggressive, red decks for players who like that playstyle and controlling blue decks for players who prefer that. This allows players to express their personality through the game.
The decision making part is also important though, and one with which standard lessons on making decisions still hold true.
Tangible Results - Giving a decision point a tangible result does a great deal to make that decision feel meaningful to a player. Seeing the chosen clothes on the avatar reminds the player of that decision as does having other players see it.
Player Effort - The effort that a player puts into a decision adds meaning to that decision. Now, in addition to whatever else it meant to the player, it also represents that effort. A good example is in Pokémon, where the player gets to choose their team of Pokémon and each one represents the effort it took to catch it.
The issue with this method of emphasizing the choice is that players may wish to express themselves in a certain way, but not be able or willing to get the necessary piece. Also, they may wish to use a piece to show off the effort it required, but may simultaneously dislike the piece for other reasons, such as aesthetics.
Reinforcement decisions - Games like Star Wars: Knights of the Old Republic ask the player to choose between the light and dark side many, many times over the course of the game. However, the game pushes you to be one or the other, and so asks you to make a choice that you have already made over and over again. Realistically, this stops becoming a decision very quickly as it has already been solved. However, it does reinforce the decision that the player has made, and so adds more meaning to it.
As a game designer, you ask a lot from your player. You ask them to abide by rules, mostly completely arbitrary, that you set down. Giving them some space to express themselves within that can do a lot to make them feel comfortable within the structure that you set down.