Fay
Glad to see it worked for you.
There are two big problems when drilling on the lathe - one is heat, and the other is keeping the hole centered. And they are related.
If you drill too fast, or as Dan mentioned, you fail to back the bit out periodically to remove the swarf, you will get friction that causes heat. And heat can destroy a blank - the excess heat causes residual moisture in the wood to flash into steam, expanding rapidly and cracking the wood.
The centering problem comes about because you are drilling wood, and wood has grain. The grain reflects the annual growth cycle of the tree - winter (slow growth = harder wood) versus summer (fast growth = softer wood). The drill bit will naturally want to follow the softer wood. Obviously, that tendency is more of a problem if the hole you are drilling is very small so that the drill bit is more flexible. I've never had a problem with a 7mm pen bit, but this is a perpetual headache with the 5/32" bit used to make '12 cent' pens using Bic refills.
The secrets to overcoming both problems are:
1. Drill far enough to fill the flutes in the bit with swarf, and then back the bit out and clean it off. How far that is depends on the size of the bit - with small bits, those flutes are very small so it's not very far - DAMHIKT.
2. Advance the bit into the wood very slowly. Allow the rotation of the wood against the bit to cut the wood ahead of its travel - if you force the bit in too fast, it is more likely to veer off course into soft grain.
3. There is a 'sweet spot' in rotational speed. Too slow and you can stall the motor. Too fast and the friction between the bit and the wood causes too much heat. The perfect speed is fast enough that the edge of the bit has a chance to cut the wood as fast - or faster - than the bit advances into the wood.
4. If you are using a small bit and are not going to be gluing something into the hole, putting a lubricant in the hole helps. I've seen recommendations for both BLO and mineral oil - I keep a block of ordinary canning paraffin next to the lathe that I used to lubricate the bit. But again - DON'T DO THIS IF YOU PLAN TO GLUE SOMETHING IN THE HOLE! Also, be aware that using a lubricant causes the swarf to clump into a hard mass, so clearing the swarf is much more important.
5. Use sharp bits. A Drill Doctor would be nice to have (I don't have one), but you can do a decent job tuning up you bits with a diamond hone.
Jim suggested using a center bit. That's a great idea, but there are things you can do that accomplish the same thing. Basically, a center bit is a heavy (inflexible) bit with a very small tip that is use to start the hole. You are effectively doing the same thing when you start the hole with an awl.
You can also drill a pilot hole. That's just a small hole that you drill first so that the final drill has a path to follow into the wood. The problem with drilling a small pilot hole is that small bits are flexible and can be diverted off center by that hard wood/soft wood issue. I find that putting the pilot hole bit as far into the chuck as possible so that I have the shortest extension of the bit minimizes the tendency to wander. Once you get that pilot hole started, the hole itself will help the bit remain on axis provided you have the rotational speed and quill advance speeds right. If you plan to drill with a brad point, forstner or spade bit, the pilot hole MUST be very small relative to the size of the tip of the bit.
The other challenge is forcing the bit to start where you want it to and not skate across the wood. Using an awl to start the hole is one solution. Center drills and pilot drills also work. Brad-point bits tend to drill where you want them to if you mark the location with an awl. The same is true of both forstner bits and spade bits.