With an 8:30 Eastern time arrival, then luggage, and rental car pick-up, you could be pushing 9:15 - 9:30.
Will your family be hungry?
I would price out one-way rentals for JAX - Cape Canaveral and back. Also length of trip rentals. Parking at the Disney PC Terminal will be about $17.50 x 8 (you pay full days, not hourly). Decide what works for you - the convenience of pulling up to the cruise and then walking right to your car afterwards (without having to stand at the rental car counter after disembarkation.) That’s a family budget decision. There are also parking lots that are less expensive that will shuttle you to/from the ship.
I can’t imagine relying on a rideshare. And I have never looked at private transfers from JAX.
I usually stay at Hilton, Marriott, and less often Hyatt properties. (Tru, Hampton Inn, Fairfield… not Hilton higher-end). Hyatt has been upping their game; don’t overlook them. Their Hyatt Place and Hyatt House hotels often provide the extra bed space that families can use - like a sofa bed. And are generally more sq ft. Best Western, Comfort Inn. I check TripAdvisor to get reviews - and also why they are scored. If it’s about bad bartenders, not bad beds, I’d book.
I would want to stay at the Southern end of the interstate loop in Jacksonville or down the interstate, just to get the metro area behind me for the morning drive in cruise day.
And I am not so concerned about being a few hours away from the port. That drive would be somewhere under 2.5 hours.
You have time to get a full-night’s sleep (Unless you are “too excited to sleep,” like the old Disney commercial). Eat breakfast and be in your way down the interstate. Know your alternate routes if something happens.
Price for hotel would definitely be a determinant of exactly where to stay.
I’d lock in a price (but definitely not pre-pay) as soon as I found something reasonable. If the price drops, you can rebook. But I’m guessing now is about the best time to book before the price rises.
We stayed at the Hyatt when we had an arrival after midnight (and free hotel shuttles to our chosen place had ended). But it would have been hot and humid while waiting for our ride - after waiting for luggage. Then paying for the ride. And then getting up in the morning to go back to the airport to get on a bus. So we took the easy way and paid for convenience and comfort. But we used companion tickets / BOGO for the flights. And got a credit on card account if spent $300 at hotel (so we had room service to bring bill up to that. I had caught a drop in room price and rebooked it. It bounced back up afterwards).
If it were the exact situation, I would consider staying at the Hyatt. But one factor different (probably the late arrival time) would have me in different (less expensive) lodgings. With less expensive transfers, too. Disney bus is not the same bubble it used to be.
Every cruise and vacation seems to be different in my life. Maybe our various ideas will help you figure out what to do for your family for this adventure. You already figured how to pivot to a different airport!