I'll expound a little on what Christian says above as, while I think his answer is correct and sufficient, it was this very same question that led to a deeper understanding of the film for me.
I also wondered about the lack of a name for the driver - until a specific moment in the film which I think explains most of the motivation for what is presented. When the driver asks his benefactor if he's familiar with the story of the scorpion and the frog, all one needs to do is look up the fable (see wikipedia) - it went a long way to answering most questions I had about the film. It also made me realize that his name was not important. I'm reminded of the Shakespeare quote "a rose by any other name would smell as sweet". Stop reading here if you don't want to see spoilers.
The driver's scorpion jacket represents his backstory (almost literally). As Christian alludes to, the title "drive" probably refers more to the character's nature than his vocation. The "kid", or driver, is by nature like the scorpion - a natural killer. The film is likely examining the questions, (1) can one's nature be suppressed? And, (2) can redemption be found by going against nature when one's nature is evil?
The scorpion (the "kid") appears to be willing to suppress his nature and replace it, for a limited time, with the nature of his victim, the frog. The car is the vehicle (sorry) by which the scorpion transforms into the frog and carrying passengers safely to their destination is the scorpion's view of a frog's nature. However, the scorpion is only willing to suppress his own nature for a limited time (5 minutes), after which, he returns to being the scorpion.
The transformation is what allows him to find love with Irene. In most scenes with her - they are either driving together or he is out of his skin (without scorpion jacket). The elevator scene (wearing jacket) is where he reveals his true nature to Irene and she is horrified by it.
The final scenes showing the kid reawakening inside the car and then driving away is perhaps a sign that he has found redemption? Maybe signifying his rebirth as the frog? Pretty cool film.