After a woman dropped a millstone on his head, why did Abimelech ask his armor-bearer to kill him?

Answer: So none could say a woman killed him

At Thebez a woman crushed his skull with an upper millstone. He said, "Kill me, lest they say of me, 'A woman killed him.'"

“But a woman dropped an upper millstone on Abimelech’s head, crushing his skull. He quickly called his armor-bearer, saying, “Draw your sword and kill me, lest they say of me, ‘A woman killed him.’” So Abimelech’s armor-bearer ran his sword through him, and he died.”— Judges 9:53-54 (BSB)