This innocent looking fellow can cause a lot of headaches. If the peg is loose in the neck, a light tap with a plastic hammer should seat it in its tapered hole. If that doesn't work, a trip to your local banjo mechanic may be in order.
Here's a maintenance tip that has saved many a peg from being replaced!
Unscrew the button:
Take off all the parts:
And place a single drop of oil on the shaft as it enters the housing:
That's the dispenser tube from my bottle of Tri-Flow, a great lubricant that's sold in bike shops. It's perfect for instrument tuners, because you can dispense one drop exactly where you want it.
Spin the shaft around with your fingers until you feel that the oil has penetrated and made it turn more smoothly. Oil the threads on the screw and reassemble the peg. It will probably work better than it has in many decades!