What's the best way to drill a hole in the end of a short spindle (maybe 8")? Mine is already turned and does not look like it would be so easy to clamp to. Is there a jig I can make to hold the spindle steady at the drill press?
If you have a lathe available (you did mention the spindle being turned already), then you have a great way to drill a hole along the center axis.
You use a drill chuck (like this one at Rockler), to hold your drill bit. You chuck the spindle, rotate and move the tailstock towards the headstock to do the drilling.
Depending on how you chuck the spindle, it may be easier to put the spindle in the headstock and the bit in the tailstock or the other way around.
I've used this technique on fairly short spindle stock, probably wouldn't work well if your spindle was a 20" long, half-inch diameter piece, as the spindle wouldn't have sufficient support.
Is there a jig I can make to hold the spindle steady at the drill press?
John Heisz has a good, short video on drilling long stock using a drill press that is worth checking out.
Basically, you swing the table of your drill press out of the way and clamp the piece to a fence that's parallel to the drill bit, then drill as usual.
Mine is already turned and does not look like it would be so easy to clamp to.
Since you have an odd piece, I would make a V-cradle, similar to the one below
and rotate it so that the cradle is upright. Then, using a combination of clamps and shims, get your spindle so that its axis is co-linear with the drill bit.
Usually, when I turn spindles that need holes in them, I drill the hole first using a Jacobs chuck in the tail stock of my lathe, then use that hole to center my cone center. That way, you're guaranteed that the hole is along the axis of the spindle and perfectly centered.
Adding to Bill Nace's answer, I recently bought a drill chuck for my lathe to do similar type drilling. Someone asked me to make wooden beads. However if you have longer items to drill into the end, you might need to add a spindle steady to the mix. that would help keep the far end from the chuck steady for the drilling.