The Blue Ridge Parkway is a great place to cycle - no trucks or buses and cars are not in a hurry. It is beautiful, but sometimes hilly. I recommend you start from the parking lot north-west of the Blue Ridge Parkway North Entrance if you are coming from Charlottesville. The route will be mostly uphill on the BRP heading out so you can more or less coast back to the home base when you turn around. You can see the elevation profile for the first 70 BRP miles on one of the images below.
The climb starts from 2000 ft at Waynesborro to 3000 ft at milepost 5 or so. You can take right turn into the the old Howardsville Turnpike at 5.9 miles (VA Rt. 610 that parallels the BRP) and cruise down and complete the loop back to your departure point. Or you can proceed for additional 5 miles on the BRP to the Humbacks Rocks Parking area and have a picnic there before returning back.
I suggest you bring a pretty good bike (road, mountain or hybrid) in working order, cycling gloves and a helmet, maybe a camera, 1 liter water and snacks. 10-15 miles total will take you one to one and half hours, depending on your conditioning.
You should also bring tools to repairs flats if needed, plus the first aid kit.Another possibility: if it is raining, if it is cold, you can stay at your home and try (I saw it in the bike shop and was amused) CycleOps Hammer.
First Aid Kit: This kit is especially designed for bikers. Sure, you can pack a few band-aids in your bag and call it a day, but the smarter cyclist is prepared for anything. First aid kits aren't just for addressing injuries, they can also give you the tools you need to keep the ride enjoyable while helping you maintain optimal form.
Bear and "Bad Guy" Horn ;8-): No really ... you will be very glad you thought this far ahead. Here's a photo of a bear that recently visited the University of Virginia! If a bear made his way to college, you can bet he crossed a few mountain roads to get there. This spray is exactly what you need to deter the bear so you can continue safely along your merry way.
Bike Pump: Don't be clueless. Get glueless! "Glueless?" you ask. Yes, the better way to deal with an air hose gone awry. Just ask the schmuck still standing there trying to patch his hose while the Tour de France is passing him by.