The Karen: As a gamer, I am able to select the gender of my avatar, and customize it to reflect my preferences of size, shape, and skin color as is appropriate to the game world.
I don't mind playing a pre-fab character, but I definitely love getting to create a hero or heroine that reflects me in some way. Escapism is my favorite part of video game play, and that starts with the ability to "own" the protagonist. Customization is a simple way to do this.
I have to say that Shepard in Mass Effect & Mass Effect 2 is the strongest example of this. My version of Shepard is a dark-haired woman with pale, lightly freckled skin and a scar on her face. I grew very attached to her through the first ME game, and when I got to resuscitate her for the second I was really excited. I had actually missed her! What's more, when I see the marketing collateral for ME I always wonder who that guy is and why they picked him. My female Shepard is vastly superior and would eat him for breakfast.
In summary: customization is a key factor for engaging this gamer. A deep game that doesn't have it is definitely missing something, and will be less likely to hold my attention (and get me to buy expansion packs, DLC, merchandise, the sequel(s), and so forth)!