My assumption is a lathe but wanted to make sure that is correct. However, I do not know what tool centers a hole in a workpiece true to the spindle.

 

Center.

Look it up here:
http://toolnotes.com/home/machining/lathes-101/lathe-workholding/

 

What is the cross section of the workpiece? Let's assume it is round; although, it could be any regular geometric figure with a center. If it is not round, then centering in a lathe chuck is more difficult.

A lathe is probably the easiest way to make a hole on center. Be sure the piece is centered in the chuck. Then use a center drill, not center, in the tailstock* to make a starting hole. After that, it depends on the accuracy you need. A regular drill will do pretty good, so long as the hole is not too deep relative to the drill's diameter. If you want it to be more accurate, I would use a boring bar after getting a rough hole made. If the lathe is fairly rigid, you can also use a center-cutting end mill.

An alternative for drilling holes at any location, including on center, is a mill.

*It is easy to just mount a drill chuck in the tailstock for holding the tool. I often use a collet for a little better accuracy/rigidity, particularly if I use an end mill.