Drilling thru a plate
Using a tile or masonry bit follow these tips:
1. Speed is more useful than pressure, so use the highest drill speed you have, or better yet, use a dremel and let the tool do the work.
2. Put the plate in a dish pan of damp sand. Shift it around until you're sure that all parts of the plate are touching the sand, so that it's evenly supported.
3. Be patient and let the tool do the work. Pushing too hard is usually what causes breakage.
4. Marking where you want to drill the hole with a piece of tape helps keep the drill bit from sliding while drilling.

Drilling a hole in a Bowling Ball
Using a masonry bit follow these tips:
1. Position the BB between 3 rocks or something else that will securely hold the ball on the ground while drilling.
2. Attach a long (at least as long as the diameter of the ball) masonry drill bit. Generally a 14" long 1/2" thick, depending on hole size needed, will do the trick.
3. Attach the drill bit to an electric drill and drill through the center of the ball by putting the drill bit down the thumb hole.
4. Take lots of breaks so that the ball and drill do not get too hot. It is safe to add a small amount of water into the hole as you drill.

Where to find the tools used above
Masonry Bit - any home improvement store