What instrument is this for? Can you share some photos?
I have no hands-on experience with zither pins, but I'd think that you'd need to use a drill bit that is made from something that has a hardness greater than the metal of the zither pin, maybe a tungsten-carbide drill bit?
Also, I'd expect that there will be tons of heat generated due to friction, and that you'd want to do this with said victim zither pin in a bench vise (and use a drill press) as opposed to leaving the zither pin on the instrument, not only for the need to prevent slight charring and ruining of the of threads in the instrument during the process, but also because the zither pin will want to rotate somewhat during the process.
I once tried to convert a cheap "1/2 size" 21" scale classical guitar to a uke bass and wanted to enlarge the existing tuner holes in the slotted-headstock tuners, and in this effort, did NOT enlarge the holes all the way thru the diameter of the tuner shaft, and ended up dulling my brass drill bits while also melting the plastic sleeve that was around the metal shaft that gives it the girth on most classical tuners. I should have done more research but this was a Sunday Midnight project and I was impatient. Possibly with the correct drill bits I would have been successful.
In the end I scrapped that project, once the Ronda Hadean uke basses were available for only $149 at launch and have not looked back to hacking up a uke bass conversion since.
Maybe you can find zither pins with a larger diameter hole, and weigh that against the cost to buy the right tool(s) and time and effort against hacking the zither pin victim?
Or, possible replacing the zither pins with alternate tuners that already have a larger hole? You can get a set of Grover 9NB geared uke tuners for ~$15 online in several places (Amazon, Sweetwater, Stewmac, Elderly), and maybe these would have large enough holes?
There are also bass guitar tuners (~$25 or less) with a smaller diameter shaft (8-10mm) that would have larger string holes than uke and/or guitar and/or dulcimer tuners in order to accommodate the thicker strings used on a bass, and some of these tuners do not have the giant elephant-ear tuner buttons that might work well.