The correct function rule that represents the total number of miles \( M \) the car will have on the odometer after \( w \) weeks is:
\[ M(w) = 121,358 + 255w \]
This means that the initial mileage of the car (121,358 miles) is added to the total mileage accumulated over \( w \) weeks (which is \( 255 \) miles per week multiplied by the number of weeks \( w \)). Therefore, the final response is:
M(w) = 121,358 + 255w