River rocks are smooth and tumbled from thousands of years of moving downstream after originating in mountains. These rocks can originate from many types of stone, some much harder than others. Some river rocks have a brittle shell that doesn't handle well for drilling. Other rocks are softer and can be drilled using the right tools.
Select several river rocks which may work for your purpose. Try to select different types of rocks based on the color, the blast sizes, texture and surface feel. In this way you can find which rocks you have more success with.
Position the rock between scrap leather in a vise. Tighten the vise so that the stone is firmly held but not so tight as to chip the stone. Your objective is to keep the stone in position while you are drilling.
Draw the size of the hole in the position you want to drill on the rock face. Roll putty into a long rope and form a circle. This needs to be large enough to fit completely around the drawn circle on the rock. Press the putty 'ring' to the rock surface until it adheres well. Add water inside the putty ring. The water is used to cool the diamond hole saw bit as you drill.
Wearing safety goggles, mask and gloves, use a heavy duty variable speed drill with a diamond hole saw bit and press firmly with the drill. Run the drill at lower speeds for the best results, higher speeds may burn out the bit.
Drill the river rock. If the bit doesn't show progress after five minutes, then try a different rock following the same process. One of the rocks should be soft enough for your hole saw to drill into. Keep the putty ring full of water as you drill.
Diamond tipped bits are used to cut holes in granite that is used on countertops and it will cut similar hard stone in its natural state. However, it may take much longer to accomplish than using the higher power tools used at a stone cutting shop.