If you're flying in and out of FLL I'd suggest getting your car there. Budget has a Fastbreak location there. If you're a member and use your Fastbreak number when you make your reservation online it is just about as easy as it gets.
After you collect your luggage you walk right out the door and into the parking garage where the car you booked or better is waiting for you.
Return is even simpler. You just pull up and the scan the car and give you a receipt right there. Check it though because getting it fixed later is a pain in the neck
Also, be sure your home auto insurance covers your car rental (most do). If so, decline their insurance every time it is offered to you either online or in person. If you ever take it once its really difficult to delete it from your profile. I learned that one the hard way.
Allow adequate time to get from FLL to the Port of Miami. Allow a minimum of 25 minutes and, depending on the time of day, as much as an hour and a half if you really get stuck in rush hour.
We paid special attention to our flight arrival times just for that reason.
I am assuming that your rental is not just for transportation to or from the port, although it might very well be the most cost effective way to go depending on your group size.
If there's just two of you a small compact car, a four door for easy luggage access, will do just fine and cost you way less than a cab or even shuttle bus. If you group is larger, consider cabs, the super shuttle or other ground transporation options
I'd give you a link to more information but I guess that is not allowed. Click on the little WWW below and look for "Start Here" then "Ground Transportation"