Good topic, Bob!
For convenience, here are the two lists from the REI site:
Updated Ten Essential "Systems"- Navigation (map and compass)
- Sun protection (sunglasses and sunscreen)
- Insulation (extra clothing)
- Illumination (headlamp/flashlight)
- First-aid supplies
- Fire (waterproof matches/lighter/candles)
- Repair kit and tools
- Nutrition (extra food)
- Hydration (extra water)
- Emergency shelter
Classic Ten Essentials- Map
- Compass
- Sunglasses and sunscreen
- Extra clothing
- Headlamp/flashlight
- First-aid supplies
- Firestarter
- Matches
- Knife
- Extra food
My own additions:
A hat is mandatory. Sunglasses are optional, unless in snow or at a beach.
Fire: If a lighter, make SURE there is a second, or backup matches.
For solo hiking off trail or off main routes: I would highly recommend carrying and using a SPOT satellite signal unit.