The proper tool for this is a metal lathe. You probably don't have one, and a drill press can work in a pinch if you've got enough throw and a good drill press vise (the kind that lets you adjust position in both dimensions would help center things up). It may be easier to get a local machine shop to do it for you if you end up having to have it made, though it won't be cheap.
Note that drilling magnets can be problematic: they're often brittle, and the hole could change the field pattern to something unsuitable. You could instead drill most of the way through the shaft then drill a hole in the "side" to avoid the magnet area, but that will make it very hard to handle potential rotation of the shaft.
There's a lot of people selling hollow joystick shafts nowadays, you may be able to find something suitable off the shelf.