We do this from Eastern PA each summer. DS is now 6 and DD is now 10.
We make the Saturday drive as far as possible and end up in Santee, SC. about 9-10 hours. We stay in and recommend the Country Inn and Suites, about $75 and includes a good breakfast buffet.
In Santee, avoid the Comfort Inn next to Cracker Barrell. Santee has two supermarkets and a few regular chains like Pizza Hut, Cracker Barrel, Bob Evans, etc. From there, Disney resort is just about 2 hours, so we are on the road by 9 Sunday morning and check in before noon after a stop at
WalMart for all the stuff we remembered leaving at home. (Happens every time, regardless of our best planning - go figure!)
Second choice in SC is Florence. Many, many choices of Hotels and Places to eat.
Saturdays in the summer can be crazy on 95 especially south of DC through northern NC. There are a few parallel roads, (Rt 1 and Rt 15) that are easy to get on and off when the traffic on 95 drives you (or the kids) crazy. Familiarize yourself with these options, you may need them.
You can't book activities until you check-in. But at check-in, go ahead and book what you want. They will fill-up in summer. A nice call about a week before to the front desk can get you a faxed copy of the weekly schedule. (Or maybe a reader here can post one?)
Wishing you safe driving!