Gubeikou (古北口) fits the bill perfectly. It's one of the easier sections to hike, it typically draws few tourists, and the surrounding landscape is spectacular. It's only a few short hours from Beijing, so if the Great Wall is on your travel radar, Gubeikou is a great section to consider visiting.
Last week, the friendly folks at The Hutong invited me to return to Gubeikou (古北口, or “Ancient North Pass”), a section of the Great Wall of China about two hours northwest of Beijing. Jeremiah and Simon, the trip's leaders, brought a wealth of knowledge of this region's history and natural setting. They did a great job of explaining the circumstances that led to the wall's construction in the 16th century. When I went to Gubeikou without a guide, I simply hiked on the wall. But this time, I could really envision Manchu armies invading from the north, and Chinese troops taking positions to defend their homeland.
Having lived in Beijing for the last three years, Katie and I have visited the Great Wall many times. Contrary to popular belief, the Great Wall isn't one continuous wall. There are many sections; some have eroded to almost nothing, others remain relatively intact, and still others that have been completely restored to their former glory. My favorite sections (as you probably guessed) are un-restored, yet still hike-able.