From the Super Mario Wiki
Thunder Sticks are items found only in Mario Kart Arcade GP, Mario Kart Arcade GP 2, and Mario Kart Arcade GP DX. The Thunder Stick is used when the item button is pressed (which starts swinging the rod). If it touches an opponent, the opponent gets zapped and knocked out briefly. It also has a limited time use. Unlike the Star, the High Beam, Hammer, 10T Hammer, and Lightning Rod, Thunder Stick does not increase speed.