My mindset is generally that it's easiest not to lug our luggage and children around more than needed.
This how we felt. LSUfan and Shmoo are both correct, overall it MIGHT not save you any time, but one way, my wife, my kids and my luggage were at the port, through security, checked in at the counter already, and waiting for me and when I got there I just went through security/check-in alone and then joined them, where as the other way, they (and the checked luggage) would have been on the shuttle with me, in the security line with me, in the check-in line with me. So for us, it was a win-win. They got to the terminal sooner (more exciting than a rental counter), and I got a little piece and quiet. Also, some people on here have reported large delays at some rental places. I wasn't delayed much by dropping off the car, but if I was, my family could have gotten on the ship when their boarding number was called, whereas if we were all together, we would all be delayed.
One thing though,
@IBLarsen, you mention a case of water. That can not be with your checked bags, so whatever adult you drop off will have to get that case of water to the port, through security, through check-in, etc. Might not be an issue if you have a little dolly for it, but I thought I'd remind you of that.