+1 for the Music Medic synthetic key oil and pivot and roller lubricant. The oil comes in a small needle applicator that is just the right size, and the "lube" comes in a handy syringe.
I am not a big fan of oiling the keys on the instrument because on the keys with long rods the oil rarely "wicks" to the center of the key where it is needed, and because it is very easy to get too much oil which runs down the post and onto the saxophone.
If a player is comfortable doing so, I recommend taking the keys off one at a time. Then you run a cotton pipe cleaner through the tubing to remove the old oil and wipe the rod with a clean cloth or paper towel. The next step is to add a drop of oil at the entrance to the hinge tube that you are going to insert the rod when the key goes back on and reinstall the key(s). One can learn to be comfortable with this method by starting with the palm keys, and side keys, and then graduate to the keys that are together on the stacks.
For the keys that are held in place by pivot screws, you remove the key (keeping the screws in the post) and clean the ends with the tip of the cotton pipe cleaner. Then you press a tiny dab of the lubricant into the holes on the ends with your fingertip, and then reattach the key. For rollers, remove the roller, clean the inside with the pipe cleaner and press a dab of lubricant into the hole, and reattach.
