If you are rigorous when you do something, you do it strictly by the book and leave nothing to chance. A rigorous inspection of your tax records has revealed the government owes you money. Congratulations!
The Latin root for rigorous is rigor, meaning “stiffness.” This might remind you or rigor mortis, the stiffening of a body after death, as often seen on cop shows. This is a good, if creepy, way of remembering the meaning of rigorous: rigorous people are stiff and unbending about the standards of their work. This requires being alive, of course.