From that TSA link:

Q. Does the name on all of my IDs have to match? What if my driver's license has only my middle initial, but my passport has my full name? Should I change my driver's license to match my passport?

A. Secure Flight does not require that the names on all of your IDs be identical. Passengers should provide their name as it appears on their government-issued ID they plan to use when traveling. This provides TSA the best information possible to use when performing watch list matching. This will result in a better process for travelers and greatly reduces the number of misidentifications. By adding date of birth and gender, the number of misidentifications is reduced further and can more readily identify passengers who do not pose a threat.

My concern would be that if the passport and BP don't match exactly, there would be some delay. Also, the guy who mentioned it to us said that his corporate travel people said that it had to do with the patriot act, but I don't know how accurate that was (and I have not looked it up.)