First - the hole doesn't have to be 'perfectly centered' in the blank. The only requirement is that there be enough wood around the tube to be able to turn the desired profile.
I don't make many kit pens, but when I do Ihave generally used a drill press to drill the blanks.
I do make a lot of '12 cent pens' and those are always drilled on teh lathe. Since the blanks that I use have a fairly small profile (typically 1/2", and sometimes even smaller), it is necessary that the hole be precise so I take lots of time. Also, the drill bit is 9/64" and tends to flex - that also means that it is necessary to be more careful. l The approach that I use is:
1. With the blank held in a four-jaw chuck, use the tip of a 'starter bit' to start the hole.
2. Move to a 'jobber-length' 9/64" bit that is held in the jacobs chuck with only about 1/2" of bit exposed for drilling to minimize the amount of flex.
3. Remount the bit to expose the full length of flute, and continue drilling
4. Switch to an 'aircraft style bit (6" long), and continue drilling. The final hole has to be 5" deep, but I back the bit out every 1" to clear the swarf. I have the lathe running at about 5oo r/m, and advance the bit into the wood very slowly so that the bit can actually cut a hole. Forcing th4e bit into the wood almust guarantees that the bit will wander with the grain in the wood.