Blood tests are being requested far more often in DWI cases. Whereas they used to be fairly rare, now it's a common occurence to request a blood sample even if a breathalyzer is available. No refusal weekends are also becoming more common - where if you refuse to take a breath test a warrant will be obtained to get a blood sample from you.
Obviously a blood test is far more invasive than a breath test. A number of people are terrified of needles, and would never willingly agree to be poked with one. It's not surprising that sometimes when an officer requests a blood sample a suspect will ask if they can take a breath test instead. It seems only fair that you should have that choice; after all, you're still giving them what they want - which is evidence they can ultimately use against you. Fairness and common sense seldom prevail in criminal cases though, and this situation is no exception.
The court's have consistently held that the choice of which test to administer belongs to the officer. They don't need a reason to choose one over the other. They get to choose, and you either have to consent to take the test they selected, or refuse. You don't get to negotiate - or make a counter-offer. Additionally, if you refuse to take the test requested that's considered a refusal, even if you offered to take the other test. That means that if they request a blood test and you tell them you'll take a breath test, that's considered a refusal.
Why would an officer request one test over another? Usually it's a matter of convenience. If you were in an accident and taken to the hospital, it's more convenient to get a blood sample - they would have to take you to jail to administer the breath test. On the other hand, if they've taken you to jail, it's more convenient to administer the breath test than have someone come in and draw blood - or to have to transport you somewhere where your blood can be drawn.
The bottom line is that you don't have a choice - either take the test requested, or refuse.