The Broken Stinger can be obtained from killing Bees in the Fields, with a rate of 1/15. It is similar to the Stinger, which can be used as a Weapon, but the tip is broken off. It can be sold to the Museum for 10,000 Coins, which is a nice source of income in the early game.
After adding the Broken Stinger to the Museum, it can be examined to show the following message:
Generally, bees will die after stinging someone due to there abdomen and digestive tract being one with the stinger. This is a rare case where only the stinger broke after this particular bee stung someone.
Too bad it's dead now.