Below is a photograph I took at the Port of Tampa just before I embarked on my cruise aboard the Carnival Inspiration. If your ship will be leaving out of Pier 2 (most likely) than this is the place:
The main thoroughfare (road) which I was standing on as I took the photograph, was Channelside Dr. You can drive your car into the pier area on the right side where you see the white taxi car and van parked in the photo. Then there should be a porter waiting to take your big checked baggage. Perhaps you could have your mother wait inside while you park the car. Then, directly off-camera, 180 degrees directly behind the camera's view, on the other side of Channelside Dr., you will see parking garages. Make sure to look for and park inside the "SHORT TERM PARKING" garage. Then go in a check-in with your mother. Ask them if it would be best to check-in with your mother or return the car first. Also, it would be better if you contact the rental car company to see if they have courtesy transportation to the pier after you drop off your car at their location.
I have been to the Port of Tampa once. I went alone and drove my own car from my home in Orlando to Tampa (which is about a 1 hour drive) and parked it in the "LONG TERM PARKING" garage.
Unfortunately, since I drove there alone, I can't tell you exactly what it is like coordinating the checking-in of your mother with the return of the rental car. But hopefully it will be just as easy. I do encourage you to ask the cruiseline, the Port of Tampa
, and the rental car company.
I hope this could be of some help to you. I also hope that you and your mother will have a great time on your cruise.