My four banjo pegs ( on the head ) as I’m restringing, I’ve noticed that at the string end,
With no string yet attached, the peg wiggles all around inside. Is this typical??
Loose peg end
I have run into that with guitar pegs. If the pegs function ok, I would think it is ok. If it is excessive (or wearing the hole) you could plug the hole and re-drill it with a closer tolerance. I did that to one of my guitars as some thought it helps the tone and that’s the way it was in pre-war guitars (I doubt it helped tone, but I felt better for trying).
I would have it checked out by a luthier