Thanks for your comments. These magnets were made by request of my sister, and ended up being quick and fun to do.
Where do you get your magnets?
I bought the 3/8" rare earth magnets from Lee Valley, along with the cup holders and friction discs. Woodcraft, etc., also sell the rare earth magnets.
Using a 4-jaw chuck to hold the rounded blank, I trued up the base and drilled a 1/2" hole using a forstner bit to fit the cup, and a 5/64" hole about 3/8" deeper, to fit a #4 x 1/2" screw. To turn the top side, I started using a jam chuck turned to fit the 1/2" hole, but didn't have enough grip, so I added a #4 screw, and made my own mini screw/jam chuck.
After assembling and inserting the magnet, I found that the magnets didn't always stay put in the cups like I expected and would stay stuck to the fridge, so I ended up epoxying them into the cup with J.B. Weld. Hindsight, I will use 1/2" magnets instead of the 3/8" magnets and cup, and just epoxy them in place.