I've drilled hundreds upon hundreds of blanks on the lathe. It's slower than a drill press but you drill very, very center on the pen.
Here is how I drill every blank, and keep in mind that I've blown out maybe 5 blanks over the past 5 years or so, so at the most. This method is a very cautious method and works for me.
This will work on any material I've done yet (many woods, tru-stone, aluminum, brass, acryic, polyester resin, "worthless wood" type blanks, etc...)
I DO NOT cut my blanks more than about 1/8" or 1/4" longer than needed. I've found it to be unnecessary once you get the hang of it, but for now, maybe a 1/2" extra is a safe way to start. You can get chipping on the back hole if you aren't careful.
1) Lathe set to 500rpm. No faster. Just causes heat buildup.
2) Grip the 4 corners of your blank with the long-nose jaws. I use a Barracuda I chuck.
3) Put your drill chuck in the tailstock. I recommend a draw bar if you have one; once in a while a dirty MT2 can cause the drill to pop out as you back out the drill... which will cause you to shart yourself.
4) Pre-drill a pilot hole using a top quality drill bit around 1/4". Advance the drill bit using your tailstock crank. Drill all the way through, but you MUST BACK THE DRILL BIT OUT every 1/4 to 3/4" (depending on the material). PURGE THE BIT OFTEN. Get a good split point bit, and my favorite is the drill bit from Woodcraft for 7mm pens or the size "O" bit from Woodcraft, both work great. For 7mm pens, you're done

.
5) Swap out the drill bit (I have two chucks; one for pre-drilling, one for final drilling, for when I make large numbers of the same pen.... moves things along very quickly).
6) Drill final hole AND MOVE VERY SLOWLY. This final cut WILL TEND TO SELF-FEED if you aren't careful. Keep tailstock locked down firmly here. The self-feeding can, with dense materials, be so strong that it can pop the drill chuck lose, causing you to shart yourself again. Hence the draw bar (which admittedly I don't use right now b/c I've learned to work around it).
I wish I could do a short video of this but it's already been done

and I don't have a digital recorder anyhow.