Mega Rush is a badge that increases Attack Power by 5 (4 in Paper Mario) when Mario is in Peril (1 HP)
In Paper Mario, it's found in Tubba Blubba's Castle. In Paper Mario: The Thousand-Year Door, it can be purchased from Charlieton for 60 coins. It uses 1 BP. The Mega Rush badges effects can be stacked up during battle and is extremely useful when dealing with stronger enemies/bosses.