The Mt. Everest or Solu Khumbu region is the second most popular trekking area in Nepal. It would probably be the most popular destination. To get near Everest, you must either walk for a week or fly to Lukla, a remote mountain airstrip where flights are notoriously unreliable.
Solu Khumbu is justifiably famous, not only for its proximity to the world's highest mountain (8,850m), but also for its Sherpa villages and monasteries. The nominal goal of an Everest trek is the Everest base camp at an elevation of about 5340m. You cannot see Everest from the base camp, so most trekkers climb Kala Pattar, a 5545m bump on the southern flank of Pumori (7145m). From Kala Pattar, there is a dramatic view of Mt. Everest.