After re-pivoting dozens of arbors without too much difficulty I guess the inevitable had to happen eventually. This page will show how to recover after breaking a small drill bit off in an arbor while drilling to install a new pivot. There are many ways to still have success after a problem like this but this seemed like the best repair in this case. The arbor I was repivoting was .073" diameter and the broken pivot needed to be replaced with a .026" pivot. I had faced off the end of the arbor smooth, touched the end with a small center drill and was drilling with a #72 high quality HSS twist drill. I had cleared chips several times and was almost to the depth I wanted when it broke off flush so no chance of pulling it out. 

This is the wheel assembly with broken pivot on the lantern pinion end. The drill bit is already broken off at this point but I don't have a picture of it. 

To hold this in the lathe to