Scott wrote ...
I need to make a specialized tool and am looking for a way to drill a
1/16" hole EXACTLY in the center of the end of a piece of say 1/2" of
3/4" solid aluminum rod. ...



Can't be done. Not "EXACTLY" anyway. If somebody hands you one
that's "exactly" dead nuts center, they simply aren't using a fine
enough means of measuring. So pick your tolerance, .005", .001",
..0005", .0001", a nanometer, an angstrom, a yoctometer, whatever.

If something within a few thousandths is sufficiently accurate, a
drill press alone will do the job. Trick is to make a one-time custom
fit vise in situ.

Take two chunks of similar dimension hardwood, say 1x2 oak. Screw
them together with a couple slips of paper between them. Don't use
bolts, only screws. Lay the assembly on the drill press table with
the seam & paper standing vertically, directly under the center of a
tiny drill bit (as close as you can eyeball it anyway). Clamp the
rear hardwood block to the drill table. Drill a 3/4" hole down
through the seam. Half of the hole will be in each piece of wood.
Leave the rear block clamped to the table while you partially unscrew
the front block from it. Remove the remnants of the paper spacer.
Put a little powdered rosin in the two halves of the hole (get a rosin
bag at a sporting goods store). Drop your aluminum stock in the hole
you just drilled & screw the front block back in place. You've now
clamped your stock in a hole that's pretty damn close to perfectly
centered under your drill chuck. Drill it to find out just how close.

Daniel