During 25 years of teaching post and rung chair making I have recommended the 5/8″ Stanley Powerbore Bit with a brace and bit extender for the horizontal boring of rung tenon mortises. The Stanley bit, with one nicker and one lip was easy to drive. The slender spear point was easy to shorten. See the Video, Make a Chair from a Tree, available through www.greenwoodworking.org Some years ago Stanley discontinued its line of Powerbore bits and I have received numerous and sometimes exasperated inquiries about a replacement. Subsequently I learned that Lee Valley manufactures and sells a bit essentially identical to the Stanley Powerbore.
However, in the interim, I discovered superior tool.. It is made in Austria by Stern and sold by Diefenbacher Tools 1-800-326-5316. Item No. 570-3510. It has two nickers and two lips. Both the nickers and the unthreaded lead point are small and require no modification. The bit is far more accurately made than the Powerbore and will hold up well. The metal is harder.The bits configuration makes it twice as aggressive as the Powerbore.
The 5/8″ Universal Bit shank diameter is .308″. I could not find a commercially available bit extender with a 5/16″ (.3125″) inside diameter socket. Brian Boggs suggested making an extender from 5/16″ diameter water hardened drill rod and a matching setscrew coupling. MSC Industrial Supply, 1-800-645-7270, Item No.06010201 and Item No. 35465731 The rod and coupling cost about 1/2 that of an extender (if you could locate one). The rod is shipped in 36″ lengths. You can make the extender any length you desire and, if necessary, the brace end of the rod can be filed to a tapered shape. I use my extender in a brace with a Universal chuck. The two-setscrew coupling works well though it is only 1″ long. File a slight flat on the both the bit shank and the drill rod.