I'm planning a trip to southern India and it sounds like I'll have a 10 hour layover from 8am to 6pm at Delhi Airport (DEL).
What sightseeing can I do during that time?
It is possible to do a few sightseeing during that amount of time. Qutub Minar is the nearest attraction from Delhi Airport (opening hours 7AM–5PM). You can reach there in a 50-60 minutes cab ride (there is no direct Metro line).
You can also also visit India Gate. Along with India Gate you can cover Parliament House, Rashtrapati Bhavan, and Connaught Place. Easiest way to reach this places will be to take Delhi Metro trains (On the Airport Express line to New Delhi Station, then on Yellow line to Central Secretariat station).
I would suggest you start your sightseeing in Connaught Place (reach by metro at Rajiv Chowk station, two station before Central Secretariat), then take a cab/auto to India Gate (Parliament House and Rashtrapati Bhavan are near to India Gate), then take Metro to Qutub Minar (Qutub Minar Station at Yellow line), then take a cab ride back to Airport.