I love games that give you the ability to customize your character. As in, create from scratch and having choices that impact directly in the game. It's something that makes a game very appealing to me. It's even better if characters react differently, depending on the customization a.k.a A town might accept you faster if you're a woman. But another town might think your character is more trustful if it's a guy, for example.
However, I've always seen games as a way to live somebody else's life. Creating my own character that has no relation whatsoever to me is awesome. But, playing as a premade character in their own story is not bad either.
I guess what I'm trying to say is that customization/immersion is something that draws me to a game. But I won't turn down a game that has a built history with a premade character.
Sometimes I feel like playing the history of the Knight that was thrown on the streets, gets back up, make connections and gets the throne. Because that's the story.
And other times I feel like creating a female with a Power Fist and a Revolver as she goes on a murdering rampage through the Wasteland. Just because I can.