A very charming, very lovely thing-- a gift to a Betty S. Clark from a L.W.R & R.M.S long ago, as written in pencil on the underside. The two certainly made a special thing from humble parts, using what looks like graphite and crayon to the create the colorful board on the lid of the small bentwood box, with a nail I assume used to make the holes and burnt match sticks cut for the pegs. I believe the game is likely a solitaire jumping game, where one eliminates pegs by jumping them, trying to wind up with one left at the center. There are just a couple short of enough pegs to fill all of the holes, but it would be very easy to add a couple of match sticks to the mix. Late-ish 19th century I believe.
3 7/8" in diameter x 1 1/16" tall and in good condition, with really nice color and a great dry wood patina.