Here is a tip for starting the drill bit into the blank when drilling on the lathe so that you get a straight hole through the blank.
Load the blank into your chuck and secure it. Then square off the end of the blank. This is the first key to a good and straight hole. If the end of the blank isn't square, the flutes of the drill bit will want to grab the higher areas, and wander off center, which will end up leaving a wallered out oblong hole.
You don't have to have a perfect finish on the end, but make sure it is fairly flush and flat.
Next, start making a dimple in the center of the blank. This is essentially a pilot hole for the drill bit to follow. This will give the drill bit a centered starting point.
I drill on the slowest speed on my lathe. The grain in a wood blank will always want to pull the drill bit off center, since there are different densities in the layers of wood. If you start out straight, you will have an excellent chance it will end up straight. If the bit starts to wander at first and you keep drilling, the hole won't be straight or it will be oblong when finished. Go slow when feeding the drill bit into the wood at first. If it starts to wobble, then pull out the drill bit and redo the dimple in the wood.
Brian